Let n be a positive odd integer. Prove that n4 ≡ 1 (mod 16). You may use any of the mod rules provided in class (In addition, if you end up getting an expression of the form a (3a+1) somewhere in your work where a is an integer, separately end up proving that this expression is always even and use that result to complete your proof).

If is odd, then we can write for some .

Expanding gives

This reduces to

Notice that is 8 times the product of two consecutive integers, one even and the other odd. This product is even and thus divisible by 2, so that where , and the equivalence further reduces to

as required.

total amount of money that he should pay if he has not used his duty free exemption is $98

step-by-step explanation:

given a united states citizen returning to the states declares the following items at the customs office: 3 shirts at $8.50 each 2 dresses at $27.50 each 1 pair of gold cuff links at $17.50 per pair. we have to find how much he pay

cost of 3 shirts if one cost at $8.50 is

cost of 2 dresses if one cost at $27.50 is  

cost of 1 pair of gold cuff at $17.50

so, total amount of money that he should pay if he has not used his duty free exemption is   $25.5+$55+$17.50=$98