ExPrEsSiOnS

Because the five numbers are consecutive, either a, c, and e are odd and b and d are even, or b and d are odd and a, c and e are even. Knowing this we can show what the values of the expressions could be. Ab+d is odd times even (even) plus either odd or even meaning the answer Is either odd or even. Ac+d is either odd times odd plus even (which is odd), or even times even plus odd (which is also odd). Ab+c is odd times even plus either odd (which is odd), or plus even (making it even). Ac+e is either odd time odd plus odd (even) or even times even plus even (even). Therefore ac+e is the only one that must be even.

Choice (A): ab+c: At least one of every two consecutive positive integers a and b must be even. Hence, the product ab is an even number.

Now, if c is odd (which happens when a is odd), ab+c must be odd. For example, if a=3,b=4, and c=5, then ab+c must equal 12+5=17, an odd number. Reject.

Choice (B): ab+d: We know that ab being the product of two consecutive numbers must be even. Hence, if d happens to be an odd number (it happens when b is odd), then the sum ab+d is also odd. For example, if a=4, then b=5,c=6 and d=7, then ab+d=3⋅5+7=15+7=23, an odd number. Reject.

Choice (C): ac+d: Suppose a is odd. Then c must also be odd, being a number 2 more than a . Hence, ac is the product of two odd numbers and must therefore be odd. Now, d is the integer following c and must be even. Hence, ac+d=odd + even = odd. For example, if a=3, then b=3+1=4,c=4+1=5 (odd) and d=5+1=6 (even) and ac+d=3×5+6=21, an odd number. Reject.

Choice (D): ac+e: Suppose a is an odd number. Then both c and e must also be odd. Now, ac is product of two odd numbers and therefore must be odd. Summing this with another odd number e yields an even number.

For example, if a=1, then c must equal 3, and e must equal 5 and ac+e must equal 1×3+5=8, an even number.

Now, suppose a is an even number. Then both c and e must also be even. Hence, ac+e= (product of two even numbers) + (an even number) = (even number) + (even number) =an even number

For example, if a=2, then c must equal 4, and e must equal 6 and the expression ac+e equals 14, an even number. Hence, in any case, ac + e is even. Correct.