Question

(i) If a three-digit number 89y is divisible by 9 find the value of y.
(ii) If 31M5 is a multiple of 9 then what are the two values obtained for M?

Answer 1

Hint: We solve this problem by first discussing the concept of divisibility rule for 9. Then we find the sum of digits of a given number in both the questions. Then as the unknown value is a single digit, we check for which values of 0 to 9, the given numbers are divisible by 9.

Complete step by step answer:
Before solving the problem let us go through the divisibility rule for number 9. Any number is said to be divisible by 9 if the sum of all the digits of the number is divisible by 9.
For example, let us take two numbers 123456 and 51795.
Now let us see the first number 123456.
Sum of its digits is 1+2+3+4+5+6 = 21. As we see 21 is not divisible by 9. So, the number 123456 is not divisible by 9.
Now let us take a look at the second number, 51795.
Sum of its digits is 5+1+7+9+5=27. As 27 is divisible by 9, the given number 51795 is divisible by 9.

(i) We are given that the number 89y is divisible by 9.
So, by using the divisibility rule of 9, the sum of the digits must be divisible by 9.
Sum of the digits is equal to $8+9+y=y+17$.
So, y+17 needs to be divisible by 9. As y is a digit it takes values from 0 to 9. Let us see which of those values satisfies our condition.
When y=0, $y+17=0+17=17$, not divisible by 9.
When y=1, $y+17=1+17=18$, is divisible by 9.
When y=2, $y+17=2+17=19$, not divisible by 9.
When y=3, $y+17=3+17=20$, not divisible by 9.
When y=4, $y+17=4+17=21$, not divisible by 9.
When y=5, $y+17=5+17=22$, not divisible by 9.
When y=6, $y+17=6+17=23$, not divisible by 9.
When y=7, $y+17=7+17=24$, not divisible by 9.
When y=8, $y+17=8+17=25$, not divisible by 9.
When y=9, $y+17=9+17=26$, not divisible by 9.
So, the sum of the digits is divisible by 9, when y=1. So, the given number 89y is divisible by 9 when y=1.
Hence, the answer is 1.

(ii) We are given that 31M5 is a multiple of 9, that is it is divisible by 9.
So, by using the divisibility rule of 9, the sum of the digits must be divisible by 9.
Sum of the digits is equal to $3+1+M+5=M+9$.
So, M+9 needs to be divisible by 9. As M+9 is divisible by 9, M also must be divisible by 9. As M is digit it takes values 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
The possible cases among them when M is divisible by 9 are if M=0 or if M=9.
So, we get two possible values of M for which the given number 31M5 is divisible by 9.
Hence the answer is M=0, 9.

Note: The main mistake that happens is one might forget to check all the possibilities of the unknown digit for which it is divisible by 9. In the first part, one should check substituting the values from 0 to 9 and should not stop if we got a value as an answer because there might be two possible values as we have got in the second part.