Multiplying Odds and Evens
Unknowns _a_, _b_, and _c_ are consecutive integers.
$$\displaystyle P = (a+b)c$$
$$\displaystyle Q = (a+b)(b+c)$$
| Quantity A | Quantity B |
|----------|----------|
| The remainder of _P_ when divided by 2 | The remainder of _Q_ when divided by 2 |
Incorrect.
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If _a_ = 4, _b_ = 5, and _c_ = 6, then
$$\displaystyle P = (4+5)(6) = (9)(6) = 54 = even$$, and
$$\displaystyle Q = (4+5)(5+6) = (9)(11) = 99 = odd$$.
In this case, Quantity A is 0, and Quantity B is 1. Therefore, Quantity A is not greater.
Incorrect.
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If _a_ = 3, _b_ = 4 and _c_ = 5, then
$$\displaystyle P = (3+4)(5) = (7)(5) = 35 = odd$$, and
$$\displaystyle Q = (3+4)(4+5) = (7)(9) = 63 = odd$$.
In this case, Quantity A and Quantity B are both 0. Therefore, Quantity B is not greater.
Incorrect.
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If _a_ = 4, _b_ = 5 and _c_ = 6, then
$$\displaystyle P = (4+5)(6) = (9)(6) = 54 = even$$, and
$$\displaystyle Q = (4+5)(5+6) = (9)(11) = 99 = odd$$.
In this case, Quantity A is 0, and Quantity B is 1. Therefore, the two quantities are not equal.
Well done!
Asking about the remainder of _P_ and _Q_ when divided by 2 is just a clever way of asking whether the numbers are odd or even.
Since _a_, _b_, and _c_ are consecutive integers, they will alternate between being even and odd. Therefore, $$(a+b)$$ is a sum of one odd and one even number, and is odd. Similarly, $$(b+c)$$ is odd by the same logic.
Therefore,
$$\displaystyle Q = (a+b)(b+c) = odd \cdot odd = odd$$.
Since _Q_ is odd, its remainder when divided by 2 will be 1, and Quantity B is 1.
However, _c_ could be odd or even, so it is not known whether _P_ is odd or even. Quantity A could be 0 or 1.
For example, if _a_ = 3, _b_ = 4, and _c_ = 5, then
$$\displaystyle P = (3+4)(5) = (7)(5) = 35 = odd$$,
so Quantity A is 1. But if _a_ = 4, _b_ = 5, and _c_ = 6, then
$$\displaystyle P = (4+5)(6) = (9)(6) = 54 = even$$,
and Quantity A is 0.
From the information given in the problem, the two quantities could be equal, or Quantity B could be greater. Therefore, the relationship cannot be determined from the information given.
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
