how come C = 88^7 mod 187 = 11 please help ?

c = 88^7 mod 187

88^7 = 40867559636992

Take 88^7 divide by 187

40867559636992 : 187 = 218543099663,058....

Take the whole part of this number --->

218543099663

Multiply this for 187

218543099663 x 187 = 40867559636981

Now subtract 88^7 and 40867559636981

40867559636992 - 40867559636981 = 11

Bye!

187 = 11 * 17.

So let's separate the problem of finding 88^7 mod 187 to the two problems

88^7 mod 11

88^7 mod 17

for the moment. 88 = 8 * 11 is a multiple of 11, hence so is 88^7. So

88^7 ≡ 0 mod 11.

As for 88^7 mod 17:

88 = 5*17 + 3, so that

88^7 ≡ 3^7 mod 17.

3^7 = (3^2)^3 * 3

→

3^7 ≡ 9^2 * (9 * 3) ≡ (81)(27) ≡ (13)(10) ≡ 130 ≡ 11 mod 17.

Modulo 187, this means the residue is simultaneously of the form

11k, for some k, 0 ≤ k ≤ 16, and

17n + 11, for some n, 0 ≤ n ≤ 10.

From this, the only solution is k=1, n=0. That is,

88^7 ≡ 11 mod 187.

Hope this helps.

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