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TOMATO Objective 21 | ISI Entrance Exam

This is a problem from TOMATO Objective 21 based on positive integers. This problem is helpful for ISI Entrance Exam. Try out the problem.

Problem: 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 $4$.

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