Find a positive inverse for 7 modulo 48. (That is, find a positive integer n such that 7n ? 1 (mod 48).)
What will be an ideal response?
Step 1 : 48 = 7· 6 + 6, and so 6 = 48 ? 7· 6.
Step 2 : 7 = 6· 1 + 1, and so 1 = 7 ? 6· 1.
Step 3 : 1 = 1· 0 + 1, and so gcd(48, 7) = 1.
Substitute back through steps 2–1:
1 = 7 ? (6· 1)= 7 ? 6 = 7 ? (48 ? 7· 6) = 7· 7 ? 48.
Thus 7· 7 ? 1 (mod 48), and so 7 is an inverse for 7 modulo 48.
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