Square modulo 7 7

Number Theory Level pending

Consider the function f : Z → Z 7 f: \mathbb Z \to \mathbb Z_7 below:

f ( n ) = n 2 m o d 7 f(n)=n^2 \mod 7

where Z 7 = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } \mathbb Z_7= \{0,1,2,3,4,5,6\} .

Find the sum of distinct values of f f .


The answer is 7.

Muhammad Rasel Parvej by Muhammad Rasel Parvej , 27, Bangladesh

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1 solution

Every integer n n is of exactly one of the forms 7 m , 7 m ± 1 , 7 m ± 2 7m, 7m\pm1, 7m\pm2 and 7 m ± 3 7m\pm3 , which implies n 2 n^2 must be one of the following forms: 7 a , 7 a + 1 , 7 a + 4 , 7 a + 2 7a, 7a+1, 7a+4, 7a+2 , respectively.

So, the answer is 0 + 1 + 4 + 2 = 7 0+1+4+2=\boxed{7} .

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