A set of cards is numbered consecutively from 1 to N. If exactly 12 of these cards have a … — ACT

A set of cards is numbered consecutively from $1$ to $N$ . If exactly $12$ of these cards have a number that is a multiple of both $6$ and $8$ , what is the greatest possible value of $N$ ?

Answer: 311

Answer

The greatest possible value of

N

311

The correct value is 311. Since the numbers must be multiples of both 6 and 8, they must be multiples of their least common multiple, which is 24. The twelfth multiple of 24 is 288, and the thirteenth multiple is 312. To have exactly 12 such multiples, the maximum number N must be at least 288 but strictly less than 312, meaning the largest integer value is 311.

Step-by-Step Solution

Find the least common multiple (LCM) of 6 and 8.

The LCM of 6 and 8 is 24.

Any number that is a multiple of both 6 and 8 must be a multiple of their least common multiple.

Calculate the 12th and 13th multiples of 24.

The 12th multiple is

12 \times 24 = 288

and the 13th multiple is

13 \times 24 = 312

To have exactly 12 multiples in the set, the set must include the 12th multiple but exclude the 13th multiple.

Determine the maximum value of

N

such that 312 is not included.

N = 311

The largest integer less than 312 is 311. If

N

were 312 or greater, the set would contain 13 or more multiples.

Key Concept

Least Common Multiple and Divisibility Properties

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