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a, b, c and d are integers. Is abcd + abc + ab + a an even

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by Max@Math Revolution » Wed Jun 19, 2019 11:53 pm

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[GMAT math practice question]

a, b, c and d are integers. Is abcd + abc + ab + a an even number?
1) abc is an odd integer
2) bcd is an odd integer
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Source: — Data Sufficiency |
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by Max@Math Revolution » Sun Jun 23, 2019 5:13 pm
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify the conditions if necessary.

Modifying the question:
For abcd + abc + ab + a = a(bcd+bc+b+1) to be even, either a must be even or bcd + bc + b + 1 must be even.

Condition 2):
If bcd is an odd integer, then b, c and d are odd integers. This implies that bc is odd, and bcd + bc + b + 1 is an even integer. Condition 2) is sufficient.

Condition 1)
If a = b = c = d = 1, then abcd + abc + ab + a = 1 + 1 + 1 + 1 = 4, which is an even integer, and the answer is 'yes'.
If a = b = c = 1 and d = 2, then abcd + abc + ab + a = 2 + 1 + 1 + 1 = 5, which is an odd integer, and the answer is 'no'.
Condition 1) is not sufficient since it doesn't yield a unique solution.

Therefore, B is the answer.
Answer: B
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