Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Idli-vada is said at common multiples of \(3\) and \(5\).
2. The first common multiple is \(15\).
3. The 10th common multiple is \(10\times15=150\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(150\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Multiples of \(3\) up to \(90\) are \(90\div3=30\).
2. Multiples of \(5\) up to \(90\) are \(90\div5=18\).
3. Common multiples are multiples of \(15\), and \(90\div15=6\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. idli \(30\), vada \(18\), idli-vada \(6\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Multiples of \(3\) up to \(900\) are \(900\div3=300\).
2. Multiples of \(5\) up to \(900\) are \(900\div5=180\).
3. Common multiples are multiples of \(15\), and \(900\div15=60\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. idli \(300\), vada \(180\), idli-vada \(60\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Multiples only of \(3\) include \(3,6,9,12,18,21,24,27,33,36,39,42,48,51,54,57\).
2. Multiples only of \(5\) include \(5,10,20,25,35,40,50,55\).
3. Common multiples are \(15,30,45,60\), so these go in the overlap.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. Yes. The overlap up to \(60\) is \(15,30,45,60\).

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. For \(2\) and \(5\): idli-only values are \(2,4,6,8,12,14,16,18,22,24,26,28,32,34,36,38,42,44,46,48,52,54,56,58\); vada-only values are \(5,15,25,35,45,55\); common values are \(10,20,30,40,50,60\).
2. For \(3\) and \(7\): idli-only values are \(3,6,9,12,15,18,24,27,30,33,36,39,45,48,51,54,57,60\); vada-only values are \(7,14,28,35,49,56\); common values are \(21,42\).
3. For \(4\) and \(6\): idli-only values are \(4,8,16,20,28,32,40,44,52,56\); vada-only values are \(6,18,30,42,54\); common values are \(12,24,36,48,60\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. Completed region lists are shown above for \((2,5)\), \((3,7)\), and \((4,6)\).

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. If the other number is \(2\), then \(2\) is the smaller idli number and \(4\) is the larger vada number; every multiple of \(4\) is also a multiple of \(2\), so there is no just-vada turn.
2. If the other number is \(8\), then \(4\) is the smaller idli number and \(8\) is the larger vada number; every multiple of \(8\) is also a multiple of \(4\), so there is no just-vada turn.
3. The other choices \(3,5,10\) have at least one larger-number multiple that is not also a multiple of the smaller number.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(2\) and \(8\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Factors of \(15\) are \(1,3,5,15\).
2. Factors of \(30\) are \(1,2,3,5,6,10,15,30\).
3. The common factors are \(1,3,5,15\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(1,3,5,15\)

Quick reading. Fill the erased diagram by first choosing headings, then testing every listed number against both headings.

1. Choose the headings multiples of \(3\) and multiples of \(4\).
2. Common multiples of \(3\) and \(4\) are multiples of \(12\), so \(12,24,48,72\) can go in the overlap.
3. Examples for only multiples of \(3\) are \(3,6,9,15,18,21\); examples for only multiples of \(4\) are \(4,8,16,20,28,32\).

This completion is valid because every side-only number belongs to exactly one heading and every overlap number belongs to both headings. Left-only \(3,6,9,15,18,21\); overlap \(12,24,48,72\); right-only \(4,8,16,20,28,32\).

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(40\times7=280\), which is below \(310\).
2. \(40\times8=320\), \(40\times9=360\), and \(40\times10=400\) are in the interval.
3. \(40\times11=440\), which is above \(410\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(320,360,400\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Multiples of \(7\) below \(40\) are \(7,14,21,28,35\); only \(35\) has digit sum \(8\).
2. A number with factors \(3\) and \(5\) is a multiple of \(15\).
3. Among such numbers below \(100\), \(45\) has digits differing by \(1\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. (a) \(35\); (b) \(45\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. The factors of \(6\) are \(1,2,3,6\).
2. Their sum is \(1+2+3+6=12\).
3. Since \(12=2\times6\), \(6\) is perfect.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(6\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. For \(20\) and \(28\), the common factors are \(1,2,4\).
2. For \(35\) and \(50\), the common factors are \(1,5\).
3. For \(4,8,12\), they are \(1,2,4\); for \(5,15,25\), they are \(1,5\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(1,2,4\); \(1,5\); \(1,2,4\); \(1,5\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Odd multiples of \(25\) are not multiples of \(50\).
2. \(25\times1=25\), \(25\times3=75\), and \(25\times5=125\).
3. None of \(25,75,125\) is divisible by \(50\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(25,75,125\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. The first idli-vada is the LCM of the two numbers.
2. \(\operatorname{lcm}(7,8)=56\), \(\operatorname{lcm}(7,9)=63\).
3. Also, \(\operatorname{lcm}(8,9)=72\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \((7,8)\), \((7,9)\), or \((8,9)\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Factors of \(28\) are \(1,2,4,7,14,28\).
2. Factors of \(70\) are \(1,2,5,7,10,14,35,70\).
3. The common factors are \(1,2,7,14\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(1,2,7,14\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. We need the largest powers \(2^3\), \(3^2\), and \(5\).
2. Multiply \(2^3\times3^2\times5=8\times9\times5\).
3. The product is \(360\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(360\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. We need \(2^3\), \(3^2\), \(5\), and \(7\).
2. Multiply \(8\times9\times5\times7\).
3. The product is \(2520\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(2520\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. The primes are \(23\) and \(29\).
2. The other numbers from \(21\) to \(30\) are composite.
3. So there are \(2\) primes and \(8\) composite numbers.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(2\) primes and \(8\) composites

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Every even number greater than \(2\) is divisible by \(2\).
2. It also has factors \(1\) and itself.
3. So it has more than two factors and is not prime.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. No. \(2\) is the only even prime.

Quick reading. The NCERT list is enough: compare neighbouring primes in order.

1. The smallest gap is between \(2\) and \(3\), so \(3-2=1\).
2. Scanning the same list, the largest jump is from \(89\) to \(97\).
3. Therefore the largest difference is \(97-89=8\).

No larger gap appears in the displayed list from \(1\) to \(100\). Smallest gap \(=1\); largest gap \(=8\).

Quick reading. Compare all ten rows of the NCERT table, not only the first and last rows.

1. The rows do not have equal counts; the table above shows the count for each decade.
2. The row \(91\) to \(100\) has only \(97\), so it has the least number of primes.
3. The rows \(1\) to \(10\) and \(11\) to \(20\) each have \(4\) primes, the most in this table.

This proves both parts of the question because every decade has been counted once. Least: \(91\)–\(100\); most: \(1\)–\(10\) and \(11\)–\(20\).

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(23\) has no factor \(2,3,\) or \(5\), so it is prime.
2. \(51=3\times17\), so it is composite.
3. \(37\) is prime, and \(26=2\times13\) is composite.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(23\) and \(37\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(2+3=5\).
2. \(3+7=10\).
3. \(2+13=15\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \((2,3)\), \((3,7)\), \((2,13)\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(13\) and \(31\) work as given.
2. \(17\) and \(71\) work, and \(37\) and \(73\) work.
3. \(79\) and \(97\) also work.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \((13,31)\), \((17,71)\), \((37,73)\), \((79,97)\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Take \(90,91,92,93,94,95,96\).
2. The even numbers in this list are composite.
3. \(91=7\times13\), \(93=3\times31\), and \(95=5\times19\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(90,91,92,93,94,95,96\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Twin primes have difference \(2\).
2. Scanning the prime list gives \((5,7),(11,13),(29,31),(41,43),(59,61),(71,73)\).
3. The two pairs in the stem are not repeated in the final list.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \((5,7),(11,13),(29,31),(41,43),(59,61),(71,73)\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. (a) True, because a number ending in \(4\) is even and greater than \(2\).
2. (b) False and (c) false, because products of primes are composite and primes have factors \(1\) and themselves.
3. (d) False because \(2\) is even and prime. (e) True because the next number after an odd prime is even and greater than \(2\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. True, False, False, False, True

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(45=3\times3\times5\), so a prime repeats.
2. \(60=2\times2\times3\times5\), so it is not exactly three distinct primes.
3. \(105=3\times5\times7\) works; \(91\) has two primes and \(330\) has four.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(105\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. If the last digit is \(2\) or \(4\), the number is even.
2. If the last digit is \(5\), the number is divisible by \(5\).
3. Every arrangement fails one of these tests.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(0\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. For \(p=5\), \(2p+1=11\).
2. For \(p=11,23,29\), the results are \(23,47,59\).
3. For \(p=41\), the result is \(83\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(p=5,11,23,29,41\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. A pair is safe only when it has no common jump size except \(1\), and jump size \(1\) is not allowed in this version of the game.
2. \(15\) and \(39\) share \(3\), so not safe; \(20\) and \(55\) share \(5\), so not safe.
3. \(4\) and \(15\) share only \(1\), and \(18\) and \(29\) share only \(1\), so these two pairs are safe.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. safe pairs are \((4,15)\) and \((18,29)\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(18=2\times3^2\) and \(35=5\times7\), so co-prime.
2. \(15=3\times5\) and \(37\) is prime, so co-prime.
3. \(30\) and \(415\) share \(5\); \(17\) and \(69\) are co-prime; \(81\) and \(18\) share \(9\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. (a), (b), and (d)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. For \(3\) and \(5\), the first common multiple is \(15=3\times5\).
2. For \(4\) and \(9\), the first common multiple is \(36=4\times9\).
3. For \(3\) and \(6\), the first common multiple is \(6\), less than \(18\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. Co-prime pairs have first common multiple equal to the product.

Quick reading. The drawing is correct when each arrow advances by the stated gap and closes when the first peg repeats.

1. For \(15\) pegs and gap \(10\), one cycle is \(1\to11\to6\to1\); rotations make the same triangular style because \(\gcd(15,10)=5\).
2. For \(10\) pegs and gap \(7\), draw \(1\to8\to5\to2\to9\to6\to3\to10\to7\to4\to1\), so all \(10\) pegs are used.
3. For \(14\) pegs and gap \(6\), draw cycles \(1\to7\to13\to5\to11\to3\to9\to1\) and \(2\to8\to14\to6\to12\to4\to10\to2\); for \(8\) pegs and gap \(3\), draw \(1\to4\to7\to2\to5\to8\to3\to6\to1\).

The gcd check explains why some pictures visit every peg and others split into shorter cycles. The rendered peg-circle diagrams above are the required thread-art pictures.

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(64=2^6\), \(104=2^3\times13\), \(105=3\times5\times7\), \(243=3^5\).
2. \(320=2^6\times5\), \(141=3\times47\), \(1728=2^6\times3^3\).
3. \(729=3^6\), \(1024=2^{10}\), \(1331=11^3\), \(1000=2^3\times5^3\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(64=2^6\), \(104=2^3\times13\), \(105=3\times5\times7\), \(243=3^5\), \(320=2^6\times5\), \(141=3\times47\), \(1728=2^6\times3^3\), \(729=3^6\), \(1024=2^{10}\), \(1331=11^3\), \(1000=2^3\times5^3\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. The number is \(2\times3\times3\times11\).
2. First \(3\times3=9\).
3. Then \(2\times9\times11=198\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(198\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Since \(1955\) ends in \(5\), divide by \(5\).
2. \(1955\div5=391\).
3. \(391=17\times23\), so \(1955=5\times17\times23\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(5,17,23\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(56\times25=(2^3\times7)\times5^2=2^3\times5^2\times7\).
2. \(108\times75=(2^2\times3^3)\times(3\times5^2)=2^2\times3^4\times5^2\).
3. \(1000\times81=(2^3\times5^3)\times3^4=2^3\times3^4\times5^3\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(2^3\times5^2\times7\); \(2^2\times3^4\times5^2\); \(2^3\times3^4\times5^3\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. For three different primes, use the smallest primes \(2,3,5\).
2. Their product is \(30\).
3. For four different primes, use \(2,3,5,7\), whose product is \(210\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(30\) and \(210\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(30=2\times3\times5\) and \(45=3^2\times5\), so no.
2. \(57=3\times19\) and \(85=5\times17\), so yes.
3. \(121=11^2\) and \(1331=11^3\), so no. \(343=7^3\) and \(216=2^3\times3^3\), so yes.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. No, Yes, No, Yes

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(225=3^2\times5^2\), but \(27=3^3\), so no.
2. \(96=2^5\times3\) contains \(24=2^3\times3\), so yes.
3. \(343=7^3\) has no \(17\), so no; \(999=3^3\times37\) has no \(11\) for \(99\), so no.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. No, Yes, No, No

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Both contain \(3\) and \(7\), so they are not co-prime.
2. The first has \(2\), which the second lacks.
3. The second has \(11\), which the first lacks, so neither divides the other.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. not co-prime; neither divides the other

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. A prime has factors \(1\) and itself.
2. Two different primes cannot share either prime as a factor.
3. So their only common factor is \(1\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. Yes, for two different prime numbers.

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Part (a) depends on the birth year; list multiples of \(4\) in that range and apply the century rule.
2. For part (b), the years are \(2024,2028,2032,\ldots,2096\).
3. The count is \((2096-2024)\div4+1=19\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. part (a) varies; part (b) has \(19\) leap years

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. A 4-digit palindrome has form \(abba\).
2. It is divisible by \(4\) when the last two digits \(ba\) are divisible by \(4\).
3. The smallest such palindrome is \(2112\) and the largest is \(8888\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. smallest \(2112\), largest \(8888\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. For even numbers, \(2+6=8\) works, but \(2+4=6\) does not.
2. For odd numbers, \(1+3=4\) works, but \(1+5=6\) does not.
3. So both statements are sometimes true.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. Sometimes true; Sometimes true

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. By \(10\), remainders are \(8,9,3,2,0,1,5\).
2. By \(5\), remainders are \(3,4,3,2,0,1,0\).
3. By \(2\), remainders are \(0,1,1,0,0,1,1\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(\begin{array}{c|ccc}78&8&3&0\\99&9&4&1\\173&3&3&1\\572&2&2&0\\980&0&0&0\\1111&1&1&1\\2345&5&0&1\end{array}\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. Checking \(8\) is enough for divisibility by \(2\) and \(4\).
2. Checking \(5\) shows the number ends in \(0\) or \(5\).
3. Here the number is even and divisible by \(5\), so it is divisible by \(10\) too.

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(5\) and \(8\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. The LCM of \(2,4,5,8,10\) is \(40\).
2. \(572\) and \(2352\) are not multiples of \(40\).
3. \(5600\), \(6000\), and \(77622160\) are multiples of \(40\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(5600,6000,77622160\)

Quick reading. First identify whether the item asks for a factor, multiple, prime test, co-prime test, or divisibility test. Then use the shortest exact rule for that type.

1. \(10000=2^4\times5^4\).
2. Group the powers as \(2^4=16\) and \(5^4=625\).
3. Then \(16\times625=10000\), and neither number ends in \(0\).

The result agrees with the original stem because each listed number has been tested directly, not guessed from a pattern alone. \(16\) and \(625\)

Quick reading. Factor the edge labels first; each cell must be a prime factor shared by its row and column.

1. Grid 1, rows \(105,20,30\) and columns \(28,125,18\): \(\begin{array}{ccc}7&5&3\\2&5&2\\2&5&3\end{array}\).
2. Grid 2, rows \(8,105,70\) and columns \(30,70,28\): \(\begin{array}{ccc}2&2&2\\3&5&7\\5&7&2\end{array}\).
3. Grid 3, rows \(63,27,190\) and columns \(45,42,171\): \(\begin{array}{ccc}3&7&3\\3&3&3\\5&2&19\end{array}\). Grid 4, rows \(343,66,44\) and columns \(28,154,231\): \(\begin{array}{ccc}7&7&7\\2&11&3\\2&2&11\end{array}\).

For example, in Grid 1 the first column gives \(7\times2\times2=28\), and the second column gives \(5\times5\times5=125\). All four grids meet the NCERT row and column products.