Question

Determine all positive number replacements less than the modulus for the question mark to make that statement true: 5+?≡1(mod7).

a) 1
b) 3
c) 6
d) 8

Answer 1

To solve the congruence 5 + ? ≡ 1 (mod 7), we find the equivalent positive number by subtracting 5 from 1 and then adding the modulus 7 to the result. The correct answer is 3, which is option b.

Step-by-step explanation:

To solve the congruence equation 5 + ? ≡ 1 (mod 7), we need to find a positive number replacement for the question mark that makes the statement true when the sum is reduced modulo 7.

We start by subtracting 5 from both sides to isolate the question mark. This gives us:

? ≡ 1 - 5 (mod 7)

? ≡ -4 (mod 7)

Since we are looking for a positive number that is less than the modulus (7), we add 7 to -4 to find the smallest positive equivalent:

? ≡ -4 + 7 (mod 7)

? ≡ 3 (mod 7)

Therefore, the replacement number for the question mark that makes the statement true is 3, which corresponds to option b.