A very easy modular arithmetic problem

Number Theory Level 2

Suppose 1 is congruent to m+n in mod m+1. What is the value on n if 0≤n≤m?

1 0 2 Any value m None m 2 \frac {m}{2} 3

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2 solutions

m + n ≡ 1 ( m o d m + 1 ) \Large{m+n \equiv 1 \pmod{m+1}} ⟹ n = 2 \Large{\implies n =2}

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We start by writing the congruence into a fraction.

m + n − 1 m + 1 \frac {m+n-1}{m+1}

So if n is any value other than 2, then the equation does not give a positive integer output.

So the answer is 2 \boxed{2}

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