Answer to Question #165644 in Combinatorics | Number Theory for Hiroshima

Question

Determine if the statement is TRUE or FALSE. Justify your answer. All numbers under discussion are

integers.

7. For each m ≥ 1 and n ≥ 1, if m is even and n is odd, then m + n2 or m2 + n is prime.

8. For n ≥ 1, if n is even or divisible by 3, then n2 + n is divisible by 6.

9. For each n ≥ 1, if n is neither even nor divisible by 3, then n2 2 1 is divisible by 6

Answer 1

CORRECTED SOLUTION

7. The statement "For each m ≥ 1 and n ≥ 1, if m is even and n is odd, then m + n² or m² + n is prime." is false. If m=2n then both m + n²=n(n+2) and m² + n=n(4n+1) are divisible by n.

8. The statement "For n ≥ 1, if n is even or divisible by 3, then n² + n is divisible by 6." is false.

If n=6k+4 then n is even, but n² + n=(36k²+12k+16)+(6k+4)=6(6k²+3k+3)+2 is not divisible by 6.

9. The statement "For each n ≥ 1, if n is neither even nor divisible by 3, then n²-1 is divisible by 6" is true.

Since n is not divisible by 3, $n=3m\pm 1$ for some integer m.

m must be even, otherwise n will be even. Let m=2k, then $n=6k\pm 1$ and

$n^2-1=(6k\pm 1)^2-1=36k^2 \pm 12k$

is divisible by 6.