Prove the following statements (using either direct or indirect proof method): state method (a) For all integers x, y, if x2(y+3) is even, then x is even or y is odd. (b) For every integer n, n is a multiple of 3 if and only if n can be expressed as the sum of 3 consecutive integers.
Prove the following statements (using either direct or indirect proof method): state method
(a) For all integers x, y, if x2(y+3) is even, then x is even or y is odd.
(b) For every integer n, n is a multiple of 3 if and only if n can be expressed as the sum of 3 consecutive integers.
(a)
For all integers x, y, if x2(y+3) is even, then x is even or y is odd.
Explanation:
We know, if a and b are integers and ab is even, then a is even or b is even.
As per this concept if x2(y+3) is even then is even or y+3 is even.
Any power of odd base is always an odd number.
Therefore, x will always be even number if is even.
If y+3 is even, then y will be odd number as 3 is odd number and the sum of two odd number is even number.
Answer: Therefore, the statement " For all integers x, y, if x2(y+3) is even, then x is even or y is odd" is true.
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