If P and Q are positive integers, and n is the decimal equivalent of PQ, which of the following must make n a finite number? I. P = 49, Q = 256 II. Q = 32 III. P = 75, Q = 384

Free Practice Question

PS · Problem Solving

655-705

Arithmetic

If $P$ and $Q$ are positive integers, and $n$ is the decimal equivalent of $\frac{P}{Q}$ , which of the following must make $n$ a finite number?
I. $P = 49, Q = 256$
II. $Q = 32$
III. $P = 75, Q = 384$

Answer Choices

Correct answer marked below

None

I only

II only

III only

I, II, III

Correct

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Comments

AvinashAug 21

Tricky one because III looks like it should work but 384 = 2^7 × 3, so there's a stray factor of 3. Nice trap.

KshitizSep 4

Yeah I fell for III initially. Forgot the denominator has to be only 2s and 5s after reducing.

SanjanaAug 26

Wait, but P = 75 isn't divisible by 3? 75 = 3 × 25, so the 3 cancels! Check again.

VedikaApr 17

I think the key is you have to reduce P/Q first before checking the denominator. II is the only one where the denominator is automatically fine regardless of P.

NirajApr 22

But does II guarantee it for every P? If Q = 32 and P is any positive integer, yeah denominator only has 2s. That seems right.

JackApr 27

Got this wrong the first time. Can someone explain why I isn't sufficient? 256 is a power of 2, so shouldn't it be finite?

ArinMay 4

256 = 2^8, and 49 isn't divisible by 2, so it stays a power of 2 in the denominator. It should work...

TiffanyMay 4

Agreed, I is fine. The issue is only III because of that factor of 3.

AlbertoSep 19

655-705 feels right, the III trap is subtle. Good practice for remembering the rule about denominators of 2 and 5 only.

What this question tests

What it tests

Your fluency with the order of operations, fractions, decimals, and basic number sense — the foundation every quant question leans on.

Common trap

Applying the order of operations out of sequence or rounding intermediate values before the final step.

If PPP and QQQ are positive integers, and nnn is the decimal equivalent of PQ\frac{P}{Q}QP​, which of the following must make nnn a finite number?I. P=49,Q=256P = 49, Q = 256P=49,Q=256II. Q=32Q = 32Q=32III. P=75,Q=384P = 75, Q = 384P=75,Q=384