Prove that there exist integers m and n so that 2m+7n=1.

31. Prove statement of Theorem : for all integers and .

Let a be an odd integer. Prove that 8|(a21).

Prove that if and are integers such that and , then either or .

Prove that if and are real numbers such that , then there exist a rational number such that . (Hint: Use Exercise 25 to obtain such that . Then choose to be the least integer such that . With these choices of and , show that and then that .) If and are positive real numbers, prove that there exist a positive integer such that . This property is called Archimedean Property of the real numbers. (Hint: If for all , then is an upper bound for the set . Use the completeness property of to arrive at a contradiction.)

20. If and are nonzero integers and is the least common multiple of and prove that.

Let (a,b)=1. Prove that (a,bn)=1 for all positive integers n.

Find the sum of the integers (a) from 1 to 35 and (b) from 1 to 2N.

Prove by induction that 1+2n3n for n1.

Prove that the statements in Exercises are true for every positive integer . 4.