GMAT Quantitative Question of the Day

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A five-section spinner, with sectors numbered 1 through 5 of equal size, is spun twice. What is the probability that the product of the two numbers obtained is divisible by 6?

$\dfrac{4}{25}$ (correct answer)
$\dfrac{6}{25}$
$\dfrac{3}{10}$
$\dfrac{7}{25}$

Explanation: When you encounter probability questions involving divisibility, think systematically about what conditions must be met. For a product to be divisible by 6, it must be divisible by both 2 and 3, since $6 = 2 \times 3$ . Let's identify which outcomes give us a product divisible by 6. With numbers 1-5 available, we need products containing at least one factor of 2 (from spinning 2 or 4) AND at least one factor of 3 (from spinning 3). Since our spinner only goes to 5, we can't get additional factors of 2 or 3 from other numbers. The favorable outcomes are:

First spin 2, second spin 3: product = 6 ✓
First spin 3, second spin 2: product = 6 ✓
First spin 4, second spin 3: product = 12 ✓
First spin 3, second spin 4: product = 12 ✓

That's 4 favorable outcomes out of

5 \times 5 = 25

total possible outcomes, giving us

\frac{4}{25}

. Choice B (

\frac{6}{25}

) likely counts outcomes where you get either a factor of 2 OR 3, but not necessarily both. Choice C (

\frac{3}{10}

) might result from miscounting favorable outcomes or total outcomes. Choice D (

\frac{7}{25}

) could come from including cases like (2,4) or (4,2), but

2 \times 4 = 8

isn't divisible by 3, so these don't work. Study tip: For divisibility problems involving composite numbers, break down the requirement into prime factors and ensure each factor appears in your product. Don't forget that divisibility by a composite like 6 requires ALL prime factors to be present.

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