Answer:

1. True

2. False

3. True

4. False

5. True

Step-by-step explanation:

To determine whether the congruences are true or false, we can check if the two numbers are equivalent (congruent) with respect to the given modulus.

1. 5 ≡ 8 (mod 3)

To check if 5 is congruent to 8 modulo 3, we divide both numbers by 3 and calculate the remainders:

5 ÷ 3 = 1 remainder 2

8 ÷ 3 = 2 remainder 2

Since both numbers have the same remainder (2) when divided by 3, this congruence is true: 5 ≡ 8 (mod 3).

2. 5 ≡ 20 (mod 4)

Dividing both numbers by 4, we get:

5 ÷ 4 = 1 remainder 1

20 ÷ 4 = 5 remainder 0

The remainders are different, so this congruence is false: 5 ≢ 20 (mod 4).

3. 21 ≡ 45 (mod 6)

Dividing both numbers by 6, we have:

21 ÷ 6 = 3 remainder 3

45 ÷ 6 = 7 remainder 3

The remainders are the same, so this congruence is true: 21 ≡ 45 (mod 6).

4. 88 ≡ 5 (mod 9)

Dividing both numbers by 9, we get:

88 ÷ 9 = 9 remainder 7

5 ÷ 9 = 0 remainder 5

These remainders are different, so this congruence is false: 88 ≢ 5 (mod 9).

5. 100 ≡ 20 (mod 8)

Dividing both numbers by 8, we have:

100 ÷ 8 = 12 remainder 4

20 ÷ 8 = 2 remainder 4

Since both remainders are the same, this congruence is true: 100 ≡ 20 (mod 8).