Question

A six digit number x3479y is divisible by 9 and 11. The possible values of (x, y) are
(1) (3, 1) (2) (8, 5) (3) (5, 8) (4) (1, 4)

Answer 1

Answer:

(3) (5,8)

Step-by-step explanation:

In order for a number to be divisible by 9, the sum of the digits has to be divisible by 9. So x + 3 + 4 +7 + 9 + y is divisible by 9. So x + y + 23 is divisible by 9. You can simplify by writing x + y + 5 is divisible by 9. (You can write 23 as 18 + 5. 18 is divisible by 9 so you can cancel it out.)

In order for a number to be divisible by 11, the difference between the sum of the alternating digits must be a multiple of 11. So (x + 4 + 9) - (3 + 7 + y) is a multiple of 11. Simplifying this would be x + 13 - 10 - y is a multiple of 11. If you simplify some more, you would get x - y + 3 is a multiple of 11.

So you have the equations x+y+5 ≡ 0 (mod 9) and x-y+3 ≡ 0 (mod 11)

That means that x+y has a remainder of 4 (or -5) when dividing by 9 and x-y has a remainder of 8 (or -3) when dividing by 11. The only answer choice that fulfills this is (3) (5,8). 5+8 divided by 9 is 1 remainder 4. 5-8 divided by 11 is 0 remainder -3 (or 8).