If a and b are both positive integers (Number Properties)

If a and b are both positive integers, is b^(a+1) - b(a^b) odd?

(1) a + (a + 4) + (a - 8) + (a + 6) + (a - 10) is odd
(2) b^3 + 3b^2 + 5b + 7 is odd

Explanation:

The trick to solving this question is to remember that for positive integers:

(a) Even^(positive integer) = even
(b) Odd^(positive integer) = odd
(c) even x even = even x odd = even
(d) even – even = even
(e) odd – odd = even
(f) even – odd = odd

Using statement (1):

5a-8 is odd
=> a is odd

Now evaluating b^(a+1) - ba^b, we get b^even -b*odd
If b is even, the expression is even^even - even = even
If b is odd, the expression is odd^even - odd = even
Therefore in either case, we can say that the expression is not odd. Sufficient.

Using statement (2):

b^3 + 3b^2 + 5b + 7 is odd
=> b^3 + 3b^2 + 5b is even
=> b is even (because if b is odd, the expression would be odd + odd + odd = odd)

Now evaluating b^(a+1) - b(a^b)
we get even^(a+1) - even*a^odd
= even - even*a^odd
= even
Therefore statement (2) is sufficient.

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