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The remainder theorem is a crucial concept in the number system that is also simple to grasp. With the use of illustrations, we will try to comprehend some intriguing notions about remainders and negative remainders. When a polynomial is divided by the other linear polynomial, the remainder theorem is used to determine the amount of remainder. This theorem states that we gain a simpler polynomial and a remainder if we divide a polynomial P[x] by a component [x-a] that isn't actually a component of the polynomial. This remaining amount is a number for P [x] at x = a, or more precisely, P [a].In essence, P[x] is divided by x-a if and only if P[a] = 0. It is used in an impressive manner to simplify polynomials of every order. The Euclidean Division of Polynomials is used in this application.
The Remainder Theorem: What is it?
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"If p[x] is any polynomial of degree bigger than or equivalent to 1 and is halved by the linear polynomial [x-a] where 'a' should be any real number, then perhaps the remainder equals p[a], declares the Remainder Theorem."
The formula of the Remainder Theorem
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The following expression provides the formula for the Remainder Theorem, p[x] = [x-a] q [x] + r [x].
r = p[a] when p[x] is partitioned by x-a,
r equals p [-b/a], whereas p [x] equals [ax+b].
Definition of Remainder Theorem
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The Remainder Theorem starts with a polynomial called p[x], wherein p[x] is a polynomial that represents one of the majors with x as its parameter.
Following that, as stated by the theory, divide that polynomial, p[x], by a linear parameter/factor, x - a, where an is only an integer. Here, a prolonged polynomial segment produces a polynomial q[x]; that is, the parameter q denotes the quotient polynomial, and the polynomial remaining is r[x]. It can be described as:
q[x] + r[x] = p[x]/x-a
Theorem of Factors/ Factor Theorem:
Identifying the roots of polynomial functions and converting them are two common applications of the factor principle. It is the remainder theorem in reverse. A synthetic division is used to resolve issues, followed by a verification for a zero remainder.
Y-x is a polynomial parameter when p(x) = 0, so if we think about it in a different way, then p(x) = 0 when y-x is a polynomial factor.
Proof of Remainder Theorem:
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A polynomial must be exhaustively divisible in order to achieve a reduced polynomial and "a" remainder of zero for the mathematics to operate in the physical world. This is probably one of the simplest approaches for determining whether the value "a" is a polynomial P root [x].
This is the result we get when we divide p[x] by x-a.
[x-a]q[x] + r[x]= p[x],
Considering that the dividend corresponds to [Divisor x Quotient] + Remainder
However, if r[x] is just the standard r [i.e.,keep in mind that when we use division by [x-a] the remainder is a fixed], then we get to the following conclusion, i.e.,
[x-a]q[x] + r = p[x]
Check out what transpires when x approaches a:
[A-A]q[A] + r = p[A]
p[a] = [0]q[a] + r
p[a] = r
Thus, it proved
Division of a Non-Zero Polynomial: Steps
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Step 1: The polynomials [dividend and divider] should be arranged from highest to lowest degree.
Step 2: To obtain the initial expression of the quotient, divide the first element of the dividend by the primary phrase of the divisor.
Step 3: To compute the remainder, multiply the divisor by the quotient's initial expression and subtract the result from the dividend.
Step 4: This balance is the dividend at this time, and the divisor will stay the very same.
Step 5: Repeat the very first step until the new dividend's degree is lower than the divisor's amount.
Difference between: [Remainder theorem and the Factor theorem]:
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Understanding the distinction between the factor and remainder theorems is essential before moving on. The primary distinctions between the factor theorem and the remainder theorem are listed below.
| Factor | Remainder | |
|---|---|---|
| 1. | According to the factor theorem, [x-a] is a factor of p[x] only if and when f[a] = 0. | According to the remainder theorem, the result of dividing p[x] by [x-a] is p[a]. |
| 2. | This method determines a linear polynomial's status as a component of the input polynomial. | It is used to locate the remainder. |
Negative Remainder Concepts:
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Negative remainder is impossible by definition, hence it doesn't exist. But for your ease, you can presume that in some circumstances. However, a negative remainder essentially suggests that in order to obtain the genuine remainder, you must add the divisor to the negative remainder.
Cyclicity in the remainder:
The characteristic of remainders known as cyclicity causes them to begin reoccurring after a particular threshold.
Table of Cyclicity:
| Cyclicity | Numbers |
|---|---|
| 1 | 1 |
| 4 | 2 |
| 4 | 3 |
| 2 | 4 |
| 1 | 5 |
| 1 | 6 |
| 4 | 7 |
| 4 | 8 |
| 2 | 9 |
| 1 | 10 |
Euler's Number's Function in Remainders:
According to Euler's Remainder Theorem, when the coprime numbers M and N are divided, the remaining is 1, or the number M raised to the Euler number of N will result in a remainder of 1. Since Euler's theorem only applies to co-prime values, it is a smart option to always double-check the digits.
Shortcuts, Tips, and Tricks of the Remainder Theorem:
- Every three-digit figure with the same components is divisible by 37.
- Five whole numbers added together in a row are always divisible by 5.
- Any mixture of nine consecutive digits has zero as the unit digit.
- Any natural digit n can be divided by 3 to get 10n-7.
- Any three consecutive natural integers can be divided by 8 to produce a product.
- Any odd number square will result in one as the remaining when divided by 8.
Illustration 1: When 4 is divided by 6, the remainder is four, not 2, but instead
Illustration 2: If you divide 5 by 6, for instance, the result is merely 5.
It is best practice to estimate the remainder in its perfect essence. To put it another way, it shouldn't be simplified.
Remainder Theorem application in the CAT exam:
Number system problems based on the remainder theorem are particularly popular among CAT students. The same holds true in that there are several ways to locate remainders, which is why, as a consequence, many theorems are applied to solve remainder-related concerns.
Past questions from CAT examinations with solutions:
Question: When [13100+17100] is divided by 25, what is the value of remainder?
Solution:
(13100 + 17100) = (15 – 2) 100 + (15 + 2) 100
Now 52 = 25, Thus, any word with a power of 5 greater than 5 and a value of 52 will be a multiple of 25. Therefore, only words with 150 or 151 should be considered when computing the remainder for the aforementioned question.
(15 - 2)100 + (15 + 2)100
150 as a coefficient is (-2)100 plus 2100
100C1 * 151* (-2)99 + 100C1 * 151* (-2)99 is the coefficient of 151.
These two words are mutually exclusive.
So, the total is zero.
2101
2 is the result of multiplying 21 by 25.
4 is the result of multiplying 22 by 25.
8 is the result of multiplying 23 by 25.
16 is the result of subtracting 24 from 25.
The remainder after multiplying 25 by 25 is 7
210 divided by 25 has a remainder of 72, 49, and -1.
Divide the remainder of 220 by 25 to get (-1)2 = 1.
Divide the remainder of 2101 by 25 to get the remainder of 2100 by 25 and the remainder of 21 by 25. This results in 1 * 2 = 2.
Question: What is the result when 123, 124, and 125 are divided by 9?
Solution:
When 123 is divided by 9, the result is -3.
The remainder produced by dividing 124 by 9 is -2.
When 123 is divided by 9, the result is -1.
The final balance is (-3) (-2)
(-1) equals -6. The necessary answer for the remainder is 3 (i.e., 9-6).
Question: A number N has 23 digits in total. N divided by 11 yields a remainder of 7. When N is divided by 33, what is the remainder?
Solution:
When a value is split by 9, the summation of its digits equals the remaining amount. Try it out. You should. Try to demonstrate it now.
Total number of digits = 23.
N/9 equals 5 as the remainder whenever N is divided by 9. Any number's remainder after dividing it by 9 equals the amount after dividing the total digits by 9. A number of the form 9k + 5 divided by 3 has a remnant of 2 (remainder of N/3 = 2).
N = 11000 + 7sN = 3m + 2sN = 11000 + 7sN = 11000 + 7 is Possible figures include 7, 29, and 51.
There are potential values of 2, 7, 5, 8, 11, 14, 20, 29, and 23.
It needs to be 33b + 29 for the value that has the form 11k + 7, as well as 3m + 2. How did we get to this conclusion?
29 is the initial natural number that meets all requirements.
Now, beginning with 29, every eleventh number has the shape of elevenK plus seven,[i.e., 11k=7] and every third digit has the form of threeM plus two, [i.e., 3m=2].
Consequently, each 33rd figure after 29 should appear on both lists
If b, k, and m are all natural integers, any value of the type 33b + 29 will also be of the form 11K + 7, as well as 3m + 2.
When the aforementioned amount is scaled by 33, 29 remains.
Question: When a value is divided by 18, the remainder is 7. When the exact value is divided by 12, n leaves the remainder. N can take how many values?
Solution: How many alternative remainders can A be divided by B, notably when we realise that dividing A by C yields a particular remainder?
The number may be 7 or 25, or 43 or 79.
When 12 is split, the remainders are 7 and 1.
N has a range of precisely two numbers.
How many numbers can it take? is the query.
The conclusion is "2," therefore.
Question: N2 divided by 24 yields a remainder of 1. What potential remainders may we obtain if we divide N by 12?
Solution: N is obviously an odd number. So, if we divide N by 24, the residue must be odd.
If the leftover after dividing N by 24 is 1, then the remainder after dividing N by N2 is also 1. We can also observe that N2 has a residual of 1 if the leftover after dividing N by 24 is -1.
N2 has a residual of 9 when N is divided by 24 with a leftover of 3.
When we divide N by 24 and the leftover is 5, N2 has a residue of 1.
When N is divided by 24 and the leftover is 7, N2 has a residue of 1.
When N is divided by 24 and the leftover is 9, N2 also has a residue of 9.
When N is divided by 24 and the leftover is 11, N2 has a residue of 1.
Therefore, when we divide N by 24, the remaining might be 5, 7, 1, or 11.
Or, if we divide N by 24, the potential remainders are 1, 5, 7, 11, 13, 17, 19, and 23.
Alternatively, 1, 5, 7, and 11 are the potential remainders when we divide N by 12.
What may we obtain as a potential remaining if we start dividing N by 12?
The response/answer is therefore "1, 5, 7, 11."
Question: When divided by 28, a prime number p that is bigger than 100 leaves a remainder, q. What number of values can q accept?
Solution: q may be 1.
A number of the pattern 28n + 2, that is a double of 2, would exist if q = 2.
Similarly, a number of the type 28n + 4—a multiple of 2—would exist when q = 4. Any number with the formula 28n+even is going to be a multi of 2.
When q = 7, a number of the type 28n + 7 that is a multi of 7 would exist.
Therefore, only one reminder that may exist are those that do not share any components with the number 28. Alternatively, figures that seem to be coprime to 28.
The number of integers that are co-prime to a particular natural number may be determined using an equation and a quicker method.
How many different values can q accept?
The answer is "12" as a result.
Question: A number divides by 14 and provides a remainder of 3, and divides by 35 and left a remainder of k. How many different values does k accept?
Solution: For illustration, suppose a count divided by 8 with a leftover of 3. When number N is divided by 24, what would be the remaining amount?
N/8 rest = 3, and N/24 rest =?
Let's examine the numbers 3, 27, 11, 35, and 43 that yield a remainder of 3 when divided by 8.
These numbers have 3, 11, 19 as remainders when divided by 24.
There might be a remainder of 3, 11, or 19.
Question: What potential remainders may there be when N is divided by 55 if N leaves a residual of 4 when divided by 33?
Solution: 33 and 55 have an LCM of 165. The beginning is here. When a value is divided by 33, it can do so with a remainder of 4, as well as 4 or 37 or 103.
The potential remainders after multiplying by 55 are 4, 15, 37, 44, 48, and 26.
What alternative remainders are there when N is divided by 55 if N leaves a remainder of 4 when split by 33?
As a result, the answer is 5.
Question: 123456789101112131415161718192021222324252627282930313233343536373839404142434481 divided by 45, what is the remainder?
Solution: You need to find Rem [1234…..434481/45].
Without a theorem to find the remainder, this looks a bit difficult. Let's take a simpler approach and break the problem down into smaller pieces.
This kind of problem becomes very easy once you understand the concept of negative remainders. Always try to reduce the dividend to 1 or -1.
45 = 9x5
Find the remainder separately and combine them later
Rem [1234…..434481/9]
The divisibility test for 9 divides the sum of the numbers by 9.
The sum in this case is 1 + 2 + 3 … 43 + 44 + 81 = 44*45 + 81
We know this is divisible by 9
= Numbers divisible by 9
= Rem [ …..434481 /9] = 0
Rem [1234…..434481/5] = 1 (only the last digit matters)
So our answer is a number that has a remainder of 1 when divided. 5 is divisible by 9.
9 multiples,
9, there is no remainder of 1 in 5. Invalid.
18, not a fifth left. invalid.
27, 1 of 5 not left. Invalid.
36, the rest is 1/5. valid. (This is our answer)
Question: What is the remainder of 123456………….4647484950 divided by 16?
Solution: To find the remainder of 2^n, just look at the last n digits.
= rem[123…484950/16]
= rem[4950/16]
= 6
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