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# TOMATO Objective 21

Suppose x & y are positive integers,x>y, and 3x+4y & 2x+3y when divided by 5,leave remainders 2&3 respectively. It follows that when (x-y) is divided by 5,the remainder necessarily equals
(a) 2 (b)1 (c) 4 (d) none of these

According to the problem,

$$(3x+2y)\equiv 2 \bmod{5}$$

$$(2x+3y)\equiv 3 \bmod{5}$$

subtracting the above 2 relations we get,

$$(x-y)\equiv (-1) \bmod{5}$$

i.e.  $$(x-y)\equiv 4 \bmod{5}$$

Hence the remainder is 4

December 22, 2013

1. The question says 3x+4y and you have solved the problem with 3x+2y (as on 6 may 2015). According to the question shouldn’t the answer be 1?

• I checked the question in my book. As per to book, in question there should be 3x+2y instead of 3x+4y. So answer is 4 and in question there is typing mistake!

2. 3x + 2y = 5p + 2, where p is an integer. ———- (1)
[Since given 3x + 2y leaves remainder 2 when divided by 5]

Similarly, 2x + 3y = 5q + 3 ——– (2)

(1) – (2): x – y = 5(p – q) – 1
==> The remainder is -1, = -1 + 5 = 4

Thus when (x – y) is divided by 5, remainder is 4.
I think this is the solution !