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Remainder Resolution from Multiple (Posted on 2016-08-04) Difficulty: 3 of 5
When a number Y is divided by X - it leaves a remainder of 11.
When Y is divided by 8X , the remainder is 92.

What is the remainder when Y is divided by 4X ?

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution Analytical solution yields 5 possible remainders | Comment 4 of 5 |
Let Y = aX + 11 = 8bX + 92

Then X = 81/(a-8b)

But 8X must be > 92, because dividing by 8X leaves a remainder of 92, so X > 11.5

Since (a-8b) is an integer, X can only be 81/n where n between 1 and 7 inclusive.  If n > 7, then X is < 11.5.
Then Y can be 92 +8m(81/n) where m = any integer.

Here are the remainders (Y mod 4X) for each of the possible n.  
The remainder is independent of m, so I only need to calculate 92 Mod (4*81/n)

n      remainder
1 92
2 92
3 92
4 11
5 27.2
6 38
7 45 + 5/7

The remainder can be any one of 5 values.  If X is an integer, then n = 1 or 3, and the remainder must by 92, as demonstrated by Charlie.

  Posted by Steve Herman on 2016-08-04 18:50:19
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