If a and b are integers, and , then cannot equal which of the following?
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This one took me a while to see. The key is that b = 17 - 2a, so 8a + b = 8a + 17 - 2a = 6a + 17. Since a is an integer, 6a + 17 has to be 17 more than a multiple of 6, i.e. it leaves remainder 5 when divided by 6.
Oh nice, that's much cleaner than how I did it. I just plugged in values for a and checked which choice was impossible.
Same here, but the mod 6 trick is way faster under time pressure.
Wait, why do we get to assume 2a + b = 17 exactly? I get that it's given, but then b depends on a, right? So isn't it just one variable essentially?
Yeah exactly, once you substitute b it becomes a single expression in terms of a. That's the whole point.
Is this really 655-705? The substitution part is easy but I keep second-guessing which answer is 'impossible.' Ended up guessing.
Don't feel bad, the trick is realizing you only need the remainder, not actual values of a and b.
Quick strategy note: whenever you see something like 'cannot equal' with integer constraints, think divisibility/remainder. Here 8a + b minus 2a + b gives 6a, a multiple of 6, so the answer must differ from 17 by a multiple of 6. That kills one choice immediately.
What it tests
Your command of divisibility, primes, factors, parity, and how the GMAT tests properties of integers.
Common trap
Forgetting that 1 is not prime, that 0 is divisible by every integer, or that negative numbers flip your assumptions.