Divisibility Check

We start by looking at the prime factorization of $72 = 2^{3} \times 3^{2}.$ It must be that $7448x24y$ is divisible by both $8$ and $9$ to be divisible by $72$ .

From here, we'll use some divisibility rules to ensure that this number is divisible by 72. In order to be divisible by 8, the number formed by the last three digits needs to be a multiple of 8. For this to happen, we need that $y$ be either 8 or 0.

For a number to be divisible by 9, the sum of its digits need to be a multiple of 9. When $y=8$ , we have that $x=8$ , which does not satisfy the given conditions. But when $y=0$ , we have that $x=7$ .
Thus, $x-y = 7 - 0= 7$ .

72= 2* 2* 2* 3 * 3

so number should be divisible by 2,8,3,9...

sum of digits=29+x+y

digit at ones place should be even.And at the same time sum should be divisible by 9.

so y=0 and x=7

So $x-y = 7.$

The number must be divisible by 8 and 9. The last 3 digits must be divisible by 8, so y = 0 or 8. If you add up the digits so far, you have 29. The sum has to be divisible by a multiple of 9. If you assume a number for y, there is always a number you can input for x so that the whole number is divisible by 9. Assuming y = 0, you know x will not equal 0, because then you still have 29, which is not divisible by 9, so y =0 has to work. So x = 7, y = 0. 7 - 0 = 7.

the number 7448x24y is divisible by 72..hence it must be divisible by both 9 and 8.. for a number to be divisible: by 9- the sum of its digits should be a multiple of 9. by 8- the last 3 digits should be a multiple of 8.

Starting with 8: 24y is a multiple of 8.(this is not 24*y). thus y can be 0 or 8..as both are the only multiples of 8 that have their hundreds and tens digits as 2 and 4... (the nos are 240 and 248)

div by 9:

case 1: with y=0 by adding the digits of the original number we get.. 29+x is a multiple of 9. therefore x=7.. as (29+7=36) is a multiple of 9.

case 2: with y=8, again by adding the digits we get x+37 is a multiple of 9. therefore x=8..as 45 is a multiple of 9. but x is not equal to y...that is when y=8, x=8 which is false acc to question.

x=7,y=0; x-y=7-0=7

Final ans:7

$74480240+1000x+y \equiv 0 \pmod {72}$ . Take away multiples of 72

$56+64x+y \equiv 0 \pmod {72}$

y must be a multiple of 8, (because all other terms and the modulus are) so either is y=0 or y=8.

First try y=0:

$56+64x\equiv 0 \pmod {72}$ . Divide by 8:

$7+8x \equiv 0 \pmod {9}$ . Take away 9x

$7-x \equiv 0 \pmod {9}$

$x=7$

Next try y=8:

$56+64x+8 \equiv 0 \pmod {72}$ . Divide by 8:

$8+8x \equiv 0 \pmod {9}$ . Take away 9x

$8-x \equiv 0 \pmod {9}$

$x=8$

But here x=y, so we exclude this answer

$x-y=7-0=\boxed{7}$

I kinda brute forced it. Took me about 20 minutes. ughh, and I know no-one will read this but I want to post what I did anyway.

So what I did was I put 1,2,3etc in for x and then I already knew 74480000 mod 72 was 32, so I carried the 32 as 32 is obviously not divisible by 72 and I just wrote down 32x. Then I input values for x, so for example 1, so 321. So 321 gives a remainder of 33 then I wrote down 33 too. Now carrying 33 to the next digit the process carries on to 334 which gives a remainder of 42 but then theres clearly no value that fits onto 42 to become divisble to 72 because the nearest multiple is 432. and I can't make y 12. So I carry on this time with x=2, it was actually not as tedious as it sounds because now I can replace my 321 with 322 and I know the next will be 34 because im just adding 1, so carrying it to the next digit you get 344 etc etc

Anyway yeah it took a while but I'm not unhappy about not doing it through prime factors.

For a number to be divisible by a composite number, it should be divisible by ALL its factors. For this problem, a number should be divisible by 72 if it is divisible by BOTH 8 and 9. Y is quite easy. Just take the last 3 digits and think of a number that would be divisible by 8. 240 looks divisible by 8 (quotient: 30). Y=0. Next, for a number to be divisible by 9, ALL digits should sum up to a multiple of 9. If we add all the digits, excluding the variables, we get 29. The closest multiple of 9 from 29 is 36 and is the ONLY multiple whose difference between 36 and 29 is a one digit number. Hence, subtracting 36 from 29 we get 7. So x=7 We're looking for x-y. Substituting the values: 7+0=7 The answer is 7