Numbers in General Form: Divisibility Test, Examples

Numbers in general form are expressed as the products of their digits with corresponding values. In addition to counting, numbers are applied for various purposes. When anything is measured, it is done so with numbers. The number system comprises a range of numerals based on convenience and requirements, for example, integers, whole and natural numbers, integers, and rational numbers, among other forms of numbers and the factors and multiples, as well as their relationships. Any positive number that we use is Natural Number (Example: 0, 1, 2, 3…).

Key Terms: Numbers, Remainder, 2-digit number, 3-digit number, Generalized form, Reversing, Digits, Divisibility

Also read: Isosceles Triangle Theorems

---

What are Numbers in General Form?

[Click Here for Sample Questions]

In Generalized Form, when a number is expressed as the product of its digits with their corresponding place values.

For 2- digits numbers:

Suppose a and b are two numbers, then the two-digit number will be ab, where a is at ten’s place and b at one’s place.

Then, in the general form, it can be written as

ab = 10 x a + b = 10a + b

If we interchange the ten’s and one’s place then,

ba = 10 x b + a = 10b + a

Note: a and b can be any digits from 1 to 9.

Example: Suppose a two-digit number 72, then

72 = 10 x 7 + 2

Here, 7 is at the ten’s place and so it is multiplied by 10. 2 is at one’s place.

If we interchange the digit the new two-digit number will be 27, then

27 = 10 x 2 + 7

Here, 2 is at the ten’s place and so it is multiplied by 10. 7 is at one’s place.

For 3- digit number:

Suppose a, b and c are three numbers, then three-digit number will be abc, where a is at hundred’s place and b at ten’s place and c at one’s place

Then, in general form it can written as

abc = 100 x a + 10 x b + c = 100a + 10b + 1 x c

If we interchange the hundred’s, ten’s and one’s place then,

bca = 100 x b + 10 x c + 1 x a = 100b + 10c + a

cab = 100 x c + 10 x a + 1 x b = 100c + 10a + b and so on

Note: a, b and c can be any digits from 1 to 9.

Example: Suppose a three digit number 543, then

540 = 100 x 5 + 10 x 4 + 1 x 0

Here, 5 is at the hundred’s place and so it is multiplied by 100. 4 is at ten’s place and it is multiplied by 10. 3 is at one’s place.

If we interchange the digit, the new three digit number will be either 435 or 345 or 354 or453, then

435 = 100 x 4 + 10 x 3 + 1 x 5

Here, 4 is at the hundred’s place and so it is multiplied by 100. 3 is at ten’s place and it is multiplied by 10. 5 is at one’s place.

Or, 354 = 100 x 3 + 10 x 5 + 1 x 4

Here, 3 is at the hundred’s place and so it is multiplied by 100. 5 is at ten’s place and it is multiplied by 10. 4 is at one’s place.

Read More:

---

Let’s Play with Numbers in General Form

[Click Here for Sample Questions]

Numbers are a lot of fun to play with, and we can learn a lot from them.

We can play by reversing the digits. Consider the following scenario with a 2-digit number:

Tony came to the conclusion that there will be no remainder after the divide. It is because the number becomes a multiple of 11 when the digits are reversed and added, there is no remainder when it is divided by 11:

The number taken is 10 a + b.

We get 10 b + a by reversing the digits.

When we add them together, we get 11a + 11 b = 11 (a + b).

As a result, we get a number that is divisible by 11.

Read More:

---

Numbers in General Form Divisibility Tests

[Click Here for Sample Questions]

A divisibility test is a quick approach to see if an integer is divisible by a specified divisor without having to divide it, usually by looking at the digits.

Divisibility by 2:

When a number is even or has an even unit digit (i.e. 0,2,4,6,8), is divisible by 2.

In generalized form,

• In the two-digit number 10a + b, the unit digit b must be any one of the numbers 0, 2, 4, 6, 8.
• In the three-digit number 100a + 10b +c, the unit digit c must be any one of the numbers 0, 2, 4, 6, 8.

Example: 22, 34, 568, 786, 100 all are divisible by 2.

Divisibility by 3 or 9:

When the sum of a number's digits is divisible by 3 or 9, the number is divisible by 3 or 9.

In generalized form,

• In two-digit number 10a + b, when (a + b) is divisible by 3 or 9, 10a + b is also divisible by 3 or 9.
• In three-digit number 100a + 10b + c, when (a + b + c) is divisible by 3 or 9, is 100a + 10b + c is also divisible by 3 or 9.

Example:

• 81, 96, 243, 729 all are divisible by 3.
• 324, 972, 90, 27 all are divisible by 9.

Divisibility by 5:

When numbers end with digits, such as 0 or 5, it is divisible by 5.

In generalized form,

• In two-digit number 10a + b, when b is 0 or 5, 10a + b is divisible by 5.
• In three-digit number 100a + 10b + c, when c is either 0 or 5, is 100a + 10b + c is divisible by 5.

Example: 45, 90, 655, 850 all are divisible by 5.

Divisibility by 10:

When a number’s unit digit is 0, it is divisible by 10.

In generalized form,

• In two-digit number 10a + b, when b is 0, 10a + b is divisible by 10.
• In three-digit number 100a + 10b + c, when c is 0, is 100a + 10b + c is divisible by 10.

Example: 50, 90, 630, 790 all are divisible by 10.

Divisibility Rules

---

Things to Remember

1. Another trick: Suppose a 2-digit number ab = 10a + b, when interchange ba = 10b + a and subtracted smaller one from larger, then

1. If a > b , then (10a + b) – (10b + a) = 9 (a - b)
2. If b > a, then (10b + a) – (10a + b) = 9 (b -a)
3. When a=b, it is zero.

In all the cases, it is seen that the number is divisible by 9 and the remainder is zero.

4. In the case of a 3 digit number, after reversing the digits and subtracting the smaller from a larger one, it can be seen that number is divisible by 11 and the remainder is zero.
5. Obtaining 3- digit numbers from Three-Digit Numbers: Pick a number with three digits. Using the picked number, create two more three-digit numbers, then simply add them. Now divide the result by 37. The remainder will be 0.
6. The general form of numbers should be always in the Indian System of Numeration and not in the International System of Numeration.

Also Read: