Page 353 - Math_Genius_V1.0_C8_Flipbook

E:\Working\Focus_Learning\Math_Genius-8\Open_Files\20_Chapter_15\Chapter_15
\ 06-Jan-2025 Surendra Prajapati Proof-6 Reader’s Sign _______________________ Date __________

(b) Sum of odd digits = b + 5, sum of even digits = 6 + 7 = 13

So, difference = b + 5 – 13 = b – 8

If b = 8, then b – 8 = 0, which is divisible by 11.

And since b is a single digit. So, the possible value of b is 8.

Example 16: If the division of a natural number N by 5 leaves a remainder 2, what could be the
ones digit of N?

Solution: When a natural number N is divided by 5 and leaves remainder 2, then N = 5K + 2, where
K is an integer.

If K = 0, N = 5 × 0 + 2 = 2 and if K = 1, N = 5 × 1 + 2 = 7

If K = 2, N = 5 × 2 + 2 = 12, ones digit is 2 and if K = 3, N = 5 × 3 + 2 = 17, ones digit is 7 and so on.
∴ The ones digit of N could be either 2 or 7.

Example 17: If N be a natural number such that N ÷ 5 leaves a remainder 3, and N ÷ 2 leaves a

remainder 1, what must be the units digit of N?
Solution: It is given that N ÷ 5 leaves a remainder 3. Therefore, units digit of N when divided by 5,

must leave a remainder 3. It is possible only when the units digit of N is either 3 or 8.

It is also given that the division of N by 2 leaves a remainder of 1. Therefore, N must be an odd
number. So, the units digit of N can be 1, 3, 5, 7 or 9. Clearly, 3 is the common value of units digit
in two cases. Thus, the required units digit of N is 3.

Practice Time 15C

1. Check the divisibility of 15288 by 3 and 9.

2. If 21a5 is a multiple of 9, what is the value of a?
3. Find the least values of ‘a’ and ‘b’ so that an even number 5aab is divisible by both 3 and 8.

4. Find the greatest value of x and y so that an odd number of the form 5yxy2x is divisible by both 3
and 5.

5. Find the smallest number which is divisible by each of 2, 3, 4, 5, 6, 8, 9 and 10.
6. Apurva says that a number is divisible by 15, if it is divisible by both 3 and 5. Is she correct? Justify.

7. A number x, when divided by 10, leaves a remainder 5. What is the ones digit of x?

8. Is there any number which is divisible by both 2 and 4 but not by 8? Give example to support your
answer.

9. If the division of a natural number N by 3 leaves a remainder of 2, what might be the ones digit of N?
10. If N be a natural number such that N ÷ 4 leaves a remainder of 3, and the division N ÷ 2 leaves a
remainder 1. What must be the units digit of N?

351 Playing With Numbers