Properties of Modular Multiplication: Congruences and Shortcuts

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

2 × 4 = 8; 8 mod 6 = 2.

Explanation

2 × 4 = 8; 8 mod 6 = 2.

About This Quiz

Patterns and Properties In Modular Multiplication - Quiz

Numbers in modular arithmetic follow special patterns! In this quiz, you’ll explore the rules and shortcuts of modular multiplication, including how congruences interact and why the “mod” step makes math cleaner. We bring you this quiz so you can see beyond the calculations and notice the underlying properties.

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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.

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)?

3² = 9; 9 mod 4 = 1.

Explanation

3² = 9; 9 mod 4 = 1.

5) Compute (15 × 18) mod 7.

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)?

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?

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

9) If a ≡ 2 (mod 9) and b ≡ 5 (mod 9), what is ab mod 9?

ab = 2 × 5 = 10. Divide 10 by 9 → remainder 1.

Explanation

ab = 2 × 5 = 10. Divide 10 by 9 → remainder 1.

10) Compute (23 m 17) mod 10.

23 × 17 = 391; 391 mod 10 = 1.

Explanation

23 × 17 = 391; 391 mod 10 = 1.

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Cierra Henderson |MBA |

K-12 Expert

Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.

Patterns and Properties in Modular Multiplication