For how many ordered pairs of positive integers (a, b), where 1 \le a \le 15 and 1 \le b \… — GMAT

For how many ordered pairs of positive integers $(a, b)$ , where $1 \le a \le 15$ and $1 \le b \le 15$ , is the value of the expression $(a^2 + a + 1)(b^3 + b + 7)$ an even integer?

Answer: 0

Answer

The expression evaluates to an odd integer for all positive integer pairs

(a, b)

, so there are 0 ordered pairs for which the value is an even integer.

The factor

(a^2 + a + 1)

simplifies to

a(a + 1) + 1

. Because

a(a + 1)

is the product of two consecutive integers, it is always even, making

a(a + 1) + 1

odd for every integer

a

. The factor

(b^3 + b + 7)

is always odd because

b^3

and

b

have matching parities (their sum is always even), so adding 7 yields an odd number. Since the product of two odd integers is always odd, the expression is never even, yielding exactly 0 ordered pairs.

Step-by-Step Solution

Examine the parity of the factor

a^2 + a + 1

a^2 + a + 1

is odd for all integer values of

a

The expression

a^2 + a = a(a + 1)

represents the product of two consecutive integers, which is always even. Adding 1 to an even integer results in an odd integer.

Examine the parity of the factor

b^3 + b + 7

b^3 + b + 7

is odd for all integer values of

b

Since

b^3

and

b

always share the same parity, their sum

b^3 + b

is always even. Adding 7 to an even integer results in an odd integer.

Determine the overall parity of the product

The product

(a^2 + a + 1)(b^3 + b + 7)

is odd for all inputs

The product of two odd integers is strictly an odd integer.

Count the number of pairs satisfying the even condition

0 pairs

Because the product is never even, zero pairs satisfy the requirement.

Key Concept

Parity rules of consecutive integer products and polynomial expressions

Estimated Time:1m 30s

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