Page 150 - Ganit Kaushal

For example:

(i) Consider 1232.
Sum of digits at odd places = 2 + 2 = 4; Sum of digits at even places = 3 + 1 = 4
Their difference = 4 – 4 = 0. So, 1232 is divisible by 11.
(ii) Consider 282216. Now, (6 + 2 + 8) – (1 + 2 + 2) = 16 – 5 = 11
Hence, the number 282216 is divisible by 11.
(iii) Consider 7282. Now, (7 + 8) – (2 + 2) = 15 – 4 = 11

Hence, the number 7282 is divisible by 11.
Some Important Properties of Divisibility

1. If a number is divisible by another number, then it is divisible by each factor of that number.
For example: 54 ÷ 18 = 3. This shows 54 is divisible by 18. Factors of 18 are 1, 2, 3, 6, 9, and
18. We can observe that 54 is also divisible by each factor of 18.
2. If a number is divisible by two co-prime numbers, then it is divisible by their product also.
For example: The number 60 is divisible by both 5 and 3. Since 5 and 3 are co-prime numbers, 60
is also divisible by the product of 5 and 3 i.e., by 5 × 3 = 15.
3. If two given numbers are divisible by a number, then their sum and difference are also divisible
by that number.
For example: The numbers 18 and 21 are divisible by 3 as 18 ÷ 3 = 6 and 21 ÷ 3 = 7. Now, their
sum, i.e., 18 + 21 = 39 is divisible by 3 as 39 ÷ 3 = 13 and their difference = 21 – 18 = 3 is also
divisible by 3.
4. If two given numbers are divisible by a number, then their product is also divisible by that
number.

For example: The numbers 18 and 21 are divisible by 3.
So, their product = 21 × 18 = 378 is also divisible by 3. ( 378 ÷ 3 = 126)
5. If there are no common prime factors between two numbers, then they are co-prime to each
other.

6. If the prime factorisation of one number is completely included in the prime factorisation of
the second number, then the second number is divisible by the first.

Solved Examples

Ex 1. Factorising 168 in several ways, justify that prime factorisation of a number is unique. (NCERT)
Sol. 168 = 2 × 84 = 2 × 2 × 42 = 2 × 2 × 2 × 21 = 2 × 2 × 2 × 3 × 7.
Again, 168 = 3 × 56 = 3 × 7 × 8 = 3 × 7 × 2 × 4 = 3 × 7 × 2 × 2 × 2 = 2 × 2 × 2 × 3 × 7.

Further, 168 = 7 × 24 = 7 × 8 × 3 = 7 × 4 × 2 × 3 = 7 × 2 × 2 × 2 × 3 = 2 × 2 × 2 × 3 × 7.
Every time we obtain the same prime factorisation except that the factors may appear in different
orders, which does not matter in the product of numbers.
Therefore, prime factorisation of a number is unique.
Ex 2. Using prime factorisation, check if the numbers 40 and 231 are co-prime or not.
Sol. Prime factorisation of 40 = 2 × 2 × 2 × 5; and prime factorisation of 231 = 3 × 7 × 11.
We observe that there are no common prime factors in the prime factorisation of both 40 and 231.
Therefore, 40 and 231 are co-prime.

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