# NonNegative Residue-Modulo

Could you please help explain these problems?:

16. Find the least non-negative residue of:

(i) 5^18 mod 11;

(ii) 4^47mod 12;

28. Show that 11 divides 10a+b if and only if 11 divide a - b. Use this to show that 11 divides 232595.

30. Find the lease non-negative residues mod 7, 11 and 13 of 58473625.

Â© BrainMass Inc. brainmass.com September 27, 2022, 4:30 pm ad1c9bdddfhttps://brainmass.com/math/number-theory/non-negative-residue-modulo-529392

## SOLUTION This solution is **FREE** courtesy of BrainMass!

** Please see the attachment for the complete solution **

16. We wish to find the following least nonnegative residues:

(please see the attached file)

(i)

(please see the attached file)

We have:

(please see the attached file)

where we used the fact that (please see the attached file), due to Fermat's little theorem, on the last step.

(please see the attached file)

(ii)

We have:

(please see the attached file)

Thus, by the Chinese remainder theorem, we have:

(please see the attached file)

28. We wish to show that 11 divides if and only if it divides and to use this fact to show that 11 divides 232595.

We have:

(please see the attached file)

whence:

(please see the attached file)

Thus we have:

(please see the attached file)

Since the last statement is true, the first statement must also be true, whence 11 divides 232595.

30. We wish to find the least nonnegative residues mod 7, 11, and 13 of 58473625.

First note that whence the least nonnegative residues mod 7, 11, and 13 of 58473625 are equal to the least nonnegative residues mod 7, 11, and 13 of (please see the attached file) Thus we first compute (please see the attached file) We have:

(please see the attached file)

Thus we have:

(please see the attached file)

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