If k is a positive integer less than 100 such that k is divisible by 6 but not divisible b… — GRE General Test

If $k$ is a positive integer less than $100$ such that $k$ is divisible by $6$ but not divisible by $8$ , what is the greatest possible value of $k$ ?

Answer: 90

Answer

The positive multiples of 6 less than 100 in descending order are 96, 90, 84, etc. The largest candidate, 96, is divisible by 8 (

96 = 8 \times 12

), so it is excluded by the conditions. The next largest candidate, 90, is divisible by 6 (

90 = 6 \times 15

) but leaves a remainder of 2 when divided by 8 (

90 = 8 \times 11 + 2

). Thus, 90 is the greatest integer satisfying all conditions.

Step-by-Step Solution

Identify the largest multiples of 6 that are less than 100.

The multiples of 6 less than 100 in descending order are 96, 90, 84, 78, ...

Finding the greatest possible value requires testing candidates starting from the largest possible multiple of 6 below 100.

Test 96 against the constraint of not being divisible by 8.

96 / 8 = 12, so 96 is divisible by 8.

The question specifies that k must NOT be divisible by 8, ruling out 96.

Test the next candidate, 90, against the constraint.

90 / 8 = 11 R 2, so 90 is not divisible by 8.

90 meets all specified criteria: it is a positive integer less than 100, divisible by 6, and not divisible by 8.

Key Concept

Properties of Multiples and Divisibility Constraints

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