if 2484X36Y is divisible by 36, find the minimum value of x-y,?

Question

please give details

Answer 1

To be honest, this question could be phrased better, but it is an interesting question if it is what I think it is. You have an 8-digit number for which six of the digits are known, and the two unknown digits are indicated by X and Y: 2484X36Y.

This number is divisible by 36. Now, 36 = 9 * 4. For a number to be divisible by 4, its last two digits must be divisible by 4. Hence, the last two digits have to be 60, 64, or 68, and Y must be 0, 4 or 8.

Also, for the number to be divisible by 9, the sum of all the digits must be divisible by 9 (the "casting out 9s" trick). Since the sum of the known digits is 27, which is already a multiple of 9, the sum of X and Y must also be a multiple of 9. Combine that with the known possible values of Y, and we have four combinations to consider:

X = 0, Y = 0
X = 9, Y = 0
X = 5, Y = 4
X = 1, Y = 8

So, checking the four possible answers, it looks like the one where X - Y is the smallest is X = 1, Y = 8, giving a value of -7.

Answer 2

Since 24840360 is a multiple of 36,
X00Y=2484X36Y-24840360
will have the same solutions.
X00Y=1000*X+Y
Since 1008 is a multiple of 36,
X00Y-1008*X=-8*X+Y
will have the same solutions.
Next
-8*X+Y is divisible by 36
implies -8*X+Y is divisible by 9,
so X-Y=-8*X+Y+9*X is divisible by 9
and since X and Y are single digits,
X+Y can equal -9,0,or 9.
The possibility X=0 and Y=9 fails because being a multiple of 36, 2484X36Y must be even so Y must be even.
X+Y=0 works because then we would have X=Y=0 and from above we know this works.
Testing the possibilities for X+Y=9, we must have Y even because 2484X36Y must be even.
Testing the possible solutions, we find the minimal X-Y attained when X=1 and Y=8.

(kudos to zanti3 for finding the correct solution first; I then went back and corrected an error in my analysis.)

Answer 3

I don't understand how you people aren't getting this question. It is an 8 digit number and you have to find what X and Y are to make it divisible by 36. Then out of all the possible combinations, subtract Y from X and see which is the smallest answer.

Well I agree on the answer of -7.

Answer 4

Is this question copied correctly? What I'm seeing shows 2484, X, 36, and Y all being multiplied together. Since there is a factor of 36 in the multiplication, it doesn't matter what X or Y is.

Now, if the question where 2484X + 36Y, it still doesn't matter because 36 will divide 2484 (the coefficient on X) and 36 will divide 36 (the coefficient on Y) and, therefore, will divide the sum no matter what X and Y are.

So, even assuming that X and Y must be positive integers, the minimum is unbounded if you let Y go to positive infinity.

Answer 5

Minus infinity
Or if negative numbers aren't allowed, then zero

It doesn't matter what x and y are, the product will always be divisible by 36 so, for example, x could be 1 and y could be 1, resulting in x-y = 0

Answer 6

2484 and 36 are both divisible by 36.

I assume your question to be (2484x)(36y)
as long as x and y are integers, the equation will be divisible by 36.

Since x and y can be any number, let x be negative infinite while y be positive infinite.

Answer of x-y is negative infinite