No. system

Number Theory Level 3

When M is divided by N the remainder is 13 , when P is divided by N remainder is 28. And when 7M+3P divided by N the remainder is 5. How many such N are possible. Where N is natural number.

3 2 5 4

1 solution

Kelly Milla
Aug 22, 2015

we can rewrite the problem into the following mathematical statements : $M \equiv 13 \pmod{N} \\ P \equiv 28 \pmod{N} \\ 7M + 3P \equiv 5 \pmod{N} \\$
Because of the 2nd equation, we can say that $N$ must be greater than $28$ .
Substitute the value of $M$ and $P$ to the 3rd equation.
$\begin{aligned} 7(13) + 3(28) & \equiv 5 \pmod{N} \\ 91 + 84 & \equiv 5 \pmod{N} \\ 175 & \equiv 5 \pmod{N} \\ 175 - 5 & \equiv 0 \pmod{N} \\ 170 & \equiv 0 \pmod{N} \end{aligned}$
Find all values of $N > 28$ such that $N$ divides $170$ .
All possible values of $N$ are the following : $34, 85, 170$
Therefore, the answer is $3$ .

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