Find a positive inverse for 7 modulo 48. (That is, ﬁnd a positive integer n such that 7n ≡ 1 (mod 48).

Name: Find a positive inverse for 7 modulo 48. (That is, ﬁnd a positive integer n such that 7n ≡ 1 (mod 48).
Brand: Quizplus
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Question

Quizplus · Accepted Answer

The Answer is Step 1:

48 = 7 \cdot 6 + 6

, and so

6 = 48 - 7 \cdot 6

.
Step 2:

7 = 6 \cdot 1 + 1

, and so

1 = 7 - 6 \cdot 1

.
Step 3:

1 = 1 \cdot 0 + 1

, and so

\operatorname { gcd } ( 48,7 ) = 1

.
Substitute back through steps

2 - 1

:

1 = 7 - ( 6 \cdot 1 ) = 7 - 6 = 7 - ( 48 - 7 \cdot 6 ) = 7 \cdot 7 - 48 ext {. }

Thus

7 \cdot 7 \equiv 1 ( \bmod 48 )

, and so 7 is an inverse for 7 modulo 48 .

Find a Positive Inverse for 7 Modulo 48