Properties of Modular Inverses: Coprimality and Computation

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.

2) Which number has no modular inverse mod 12?

5

7

6

11

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

4

9

10

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

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.

5) Find the modular inverse of 7 mod 26.

3

5

15

19

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?

2

7

18

21

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?

3

6

10

14

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?

2

3

13

15

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?

4

5

8

15

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

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.