Properties of Modular Multiplication: Congruences and Shortcuts

1) If a ≡ 4 (mod 6), what is 2a (mod 6)?

2

4

6

8

2 × 4 = 8; 8 mod 6 = 2.

Explanation

2 × 4 = 8; 8 mod 6 = 2.

2) Which pair is not congruent (mod 7)?

3 × 5 and 2 × 4

6 × 2 and 1 × 5

4 × 6 and 2 × 5

7 × 3 and 5 × 5

7×3 = 21 ≡ 0 (mod 7), 5×5 = 25 ≡ 4 (mod 7). Not equal, so not congruent.

Explanation

7×3 = 21 ≡ 0 (mod 7), 5×5 = 25 ≡ 4 (mod 7). Not equal, so not congruent.

3) Compute (7 × 11) mod 9.

2

5

6

7

7 × 11 = 77; 77 mod 9 = 5.

Explanation

7 × 11 = 77; 77 mod 9 = 5.

4) If a ≡ 3 (mod 4), what is a² (mod 4)?

0

1

2

3

3² = 9; 9 mod 4 = 1.

Explanation

3² = 9; 9 mod 4 = 1.

5) Compute (15 × 18) mod 7.

0

2

4

6

15 × 18 = 270; 270 mod 7 = 4.

Explanation

15 × 18 = 270; 270 mod 7 = 4.

6) If a ≡ 2 (mod 5), what is a × 3 (mod 5)?

0

1

2

3

2 × 3 = 6; 6 mod 5 = 1.

Explanation

2 × 3 = 6; 6 mod 5 = 1.

7) If 8 × x ≡ 4 (mod 12), what is one solution for x?

1

2

3

4

8 × 2 = 16; 16 mod 12 = 4.

Explanation

8 × 2 = 16; 16 mod 12 = 4.

8) Which of the following is true?

Modular multiplication always increases numbers

(a × b) mod n = ((a mod n) × (b mod n)) mod n

Modular multiplication never equals zero

Multiplication modulo n always gives odd results

Option b accurately represents a fundamental property of modular arithmetic known as the distributive property. This property states that the product of two numbers, taken modulo n, is equivalent to first taking each number modulo n, multiplying them, and then taking that product modulo n. The other options are incorrect as they either misrepresent the nature of modular multiplication or state false properties about it.

Explanation

Option b accurately represents a fundamental property of modular arithmetic known as the distributive property. This property states that the product of two numbers, taken modulo n, is equivalent to first taking each number modulo n, multiplying them, and then taking that product modulo n. The other options are incorrect as they either misrepresent the nature of modular multiplication or state false properties about it.