22. Prove: if a, b, and c are odd, and a | b - c and a bc, then a | b and a c 22. Prove: if a, b, and c are odd, and a | b - c and a bc, then a | b and a c
13. Prove that for all integers b, if b is odd then b is odd 13. Prove that for all integers b, if b is odd then b is odd
Prove that if a and b are odd then 2gcd(a,b)=gcd(a+b,a−b)”
6. Prove that if a and b are odd integers, then a2 is divisible by 8. 7. Prove that if a is an odd integer, then ta + (a + 2)?+ (a +4)2 +1) is divisible by 12.
b. Prove that if a is an odd integer, then a | b2 – 1 implies that a = (a, b – 1)(a, b + 1).
8.(10 pts) PROVE by contrapositive: If c is an odd integer, then the equation n2 + n c = 0 has no integer solution for n. 8. (10 pts) PROVE by contrapositive: If c is an odd integer, then the equation na +n -c=0 has no integer solution for n.
(b) Prove that n is an odd pseudoprime number if and only if 2"-1-1 mod n. (b) Prove that n is an odd pseudoprime number if and only if 2"-1-1 mod n.
Prove that there do not exist prime numbers a, b, and c such that a^3 + b^3= c^3. prove it by using the 4 cases, use a correct and complete English sentence. 1) a,b,c are all even if a,b,c even, a^3 + b^3 = c^3 2) If a, b, c are all odd 3) a is even and b is odd 4) c even, a and b odd if a and b odd, then a^3 + b^3 even
Let A, B, and C be three collinear points s.t. A*B*C. Prove each of the follow set equalities. I'm really having trouble applying theorems like the ruler placement postulate or betweenness theorem to help prove these. 24. Let A, B, and C be three collinear points such that A * B * C. Prove each of the following set equalities. (a) BÁ U BỎ "АС (b) BA n BC {B} (c) ABU BC AC (d) AB n BC {B} (e)...
Assume n is an integer. Prove that n is odd iff 3n2 + 4 is odd. Remember that to prove p iff q, you need to prove (i) p → q, and (ii) q → p. Use the fact that any odd n can be expressed as 2k + 1 and any even n can be expressed as 2k, where k is an integer. No other assumptions should be made.