Question

Decide whether each of these integers is congruent to 3 modulo 7.
(a) 37
(b) 66
(c) -17
(d) -67

Answer 1

Solution. If a and b any integer and m is a positive integer then a congruent to b modulo m when m is divides the difference a-b. Therefore

(a)

$$\frac{37-3}{7}=\frac{34}{7}$$

37 is not congruent to 3 modulo 7.

(b)

$$\frac{66-3}{7}=\frac{63}{7}=9$$

66 is congruent to 3 modulo 7.

(c)

$$\frac{-17-3}{7}=\frac{-20}{7}$$

-17 is not congruent to 3 modulo 7.

(d)

$$\frac{-67-3}{7}=\frac{-70}{7}=-10$$

-67 is congruent to 3 modulo 7.

Answer. (a) 37 is not congruent to 3 modulo 7; (b) 66 is congruent to 3 modulo 7; (c) -17 is not congruent to 3 modulo 7; (d) -67 is congruent to 3 modulo 7.