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Example 11: Which of the following numbers is divisible by 2?

(a) 2029 (b) 4086 (c) 3123 (d) 14357
Solution: We know that a number is divisible by 2 only if its ones digit is even.
So, option (b), i.e. 4086, is divisible by 2.

Example 12: Check divisibility of the following numbers by 9 as well as 3.
(a) 8235 (b) 9061 (c) 5712 (d) 8631

Solution: (a) Given number is 8235.
∴ Sum of digits = 8 + 2 + 3 + 5 = 18, which is divisible 9. Remember
Hence, 8235 is divisible by 9 as well as 3. Remember: If a number
x is divisible by another
(b) Given number is 9061. number y, then it is also
∴ Sum of digits = 9 + 0 + 6 + 1 = 16, which is not a multiple of divisible by each of the
9 and 3. factors of the number y.
Hence, 9061 is not divisible by 9 and 3.

(c) Given number is 5712.
∴ Sum of digits = 5 + 7 + 1 + 2 = 15, which is not divisible by 9 but divisible by 3.
So, 5712 is divisible by 3, but not by 9.

(d) Given number is 8631.
∴ Sum of digits = 8 + 6 + 3 + 1 = 18, which is divisible by both 9 and 3. Hence, 8631 is
divisible by both 9 and 3.

Example 13: If 31y5 is a multiple of 3, where y is an even digit, what could be the value of y?
Solution: Sum of the digits of number 31y5 = 3 + 1 + y + 5 = 9 + y
The number is divisible by 3 if the sum of its digit, i.e., (9 + y) is divisible by 3.

This is only possible when y = 0, 3, 6 or 9. But, since y is an even digit, the value of y could be 0 and 6.
Example 14: If 30z6 is a multiple of 9, where z is an odd digit, what is the value of z?

Solution: Sum of the digits of number 30z6 = 3 + 0 + z + 6 = 9 + z
Hence, for the number to be divisible by 9, possible value of z is either 0 or 9.
But, since z is an odd digit. Hence, the value of z = 9.

Example 15: For what value of b, the following numbers are divisible by 11?
(a) 23b (b) 6b75

Solution: We know that the difference between the sum of odd digits and the sum of even digits
should be either 0 or a multiple of 11 for a number to be divisible by 11.

(a) Sum of odd digits of the number 23b = 2 + b, sum of even digits = 3
So, difference = 2 + b –3 = b – 1
If b = 1, then difference = 1 – 1 = 0, which is divisible by 11.

If b = 12, then difference = 12 – 1 = 11, which is divisible by 11.
But, the value of b cannot be 12 as b is a single digit.

∴ The value of b is 1.

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