# Modular multiplication

How about multiplication?
If you multiply *n* by 5 on a clock-face,
is there an inverse which undoes this?
Yes: multiply by 5 again!
Try it: (4×5)×5 = 4 (mod 12).
Try it again: (9×5)×5 = 9 (mod 12).

Why does this work? Consider that multiplying by 5 twice is multiplying by 25. Multiplying by 25 is an identity on the 12-hour clock-face: it does nothing. Similarly, multiplying by 7 undoes itself, because ×49 is an identity, and ×11 undoes itself, because ×121 is an identity.

1, 25, 49, 121 - all these numbers are ≡ 1 (mod 12).
25 = 2×12+1, 49 = 4×12+1, et cetera.
So, ×*a* is the inverse of ×b if *ab* ≡ 1.
Since *ab* ≡ *ba*, these inverses are paired up.

It’s coincidence that the inverses are squares for mod-12. For mod-10, ×3 and ×7 are inverses, because 3×7 ≡ 1 (mod 10).

If the modulus is *pq*, what inverses exist?
For example, if *p* = 7 and *q* = 5,

i.e. if *ab* = *kn* + 1.

What numbers are congruent to 1 in mod-12? 1, 13, 25, 37,

3

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