Properties of Modular Inverses: Coprimality and Computation

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1) If a × b ≡ 1 (mod n), then b is:

A multiple of a

A divisor of n

The modular inverse of a mod n

Always equal to n – a

By definition, b is the modular inverse of a mod n.

Explanation

By definition, b is the modular inverse of a mod n.

About This Quiz

Modular inverses follow fascinating rules based on number properties. In this quiz, you’ll discover the conditions for an inverse to exist, how greatest common divisors (GCDs) play a role, and why every nonzero number has an inverse modulo a prime. We bring you this quiz to help you see the... see morestructure behind modular systems.
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2) What first name or nickname would you like us to use?

You may optionally provide this to label your report, leaderboard, or certificate.

2) Which number has no modular inverse mod 12?

gcd(6,12) = 6 ≠ 1 → no inverse.

Explanation

gcd(6,12) = 6 ≠ 1 → no inverse.

3) If 3 × x ≡ 1 (mod 13), what is x?

3 × 9 = 27; 27 mod 13 = 1. So x = 9.

Explanation

3 × 9 = 27; 27 mod 13 = 1. So x = 9.

4) Which statement about modular inverses is true?

Every integer has an inverse modulo n

Only prime numbers have inverses modulo n

A number has an inverse mod n if gcd(number, n) = 1

Inverses are always odd

A number has a modular inverse with respect to n if it is relatively prime to n, meaning their greatest common divisor (gcd) is 1. Therefore, option C is the correct statement regarding modular inverses.

Explanation

5) Find the modular inverse of 7 mod 26.

7 × 15 = 105; 105 mod 26 = 1. So the inverse is 15.

Explanation

7 × 15 = 105; 105 mod 26 = 1. So the inverse is 15.

6) If 9 × x ≡ 1 (mod 23), what is x?

9 × 18 = 162; 162 mod 23 = 1. So x = 18.

Explanation

9 × 18 = 162; 162 mod 23 = 1. So x = 18.

7) Which of these numbers has an inverse mod 21?

gcd(10,21) = 1 → 10 has an inverse.

Explanation

gcd(10,21) = 1 → 10 has an inverse.

8) Which number is the modular inverse of 4 mod 17?

4 × 13 = 52; 52 mod 17 = 1. So the inverse is 13.

Explanation

4 × 13 = 52; 52 mod 17 = 1. So the inverse is 13.

9) If 5 × y ≡ 1 (mod 19), what is y?

5 × 4 = 20; 20 mod 19 = 1. So y = 4.

Explanation

5 × 4 = 20; 20 mod 19 = 1. So y = 4.

10) Which of these is true about inverses mod a prime number p?

Only 1 and p−1 have inverses

Every nonzero number mod p has an inverse

No number has an inverse

Only prime numbers < p have inverses

In modular arithmetic, particularly with a prime modulus p, every nonzero integer less than p has a unique multiplicative inverse. This is because a prime number has no divisors other than 1 and itself, ensuring that the equation ax ≡ 1 (mod p) has a solution for every a in the range 1 to p-1.

Explanation

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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.