Question

The cube of an odd integer is always an odd number

Answer 1

$(2n+1)^3=8 n^3 + 12 n^2 + 6 n + 1$

The first three terms are even numbers for any integer $n$. So is their sum. 1 is an odd number. The sum of an even number and an odd number is an odd number. Therefore the statement from the question is true.