A school band has between 100 and 150 members. When the band members line up in rows of 6,… — ACT

A school band has between $100$ and $150$ members. When the band members line up in rows of $6$ , there are $2$ members left over. When they line up in rows of $9$ , there are also $2$ members left over. When they line up in rows of $5$ , there are no members left over. How many members are in the school band?

Answer: 110 members

Answer

There are 110 members in the school band.

The number of members minus 2 must be a multiple of the least common multiple of 6 and 9, which is 18. The possible values between 100 and 150 are 110, 128, and 146. Among these, only 110 is divisible by 5, which satisfies the condition of having no members left over when grouped in rows of 5.

Step-by-Step Solution

Find the least common multiple (LCM) of the row sizes 6 and 9.

The LCM of 6 and 9 is 18.

Since the remainder is the same (2) for both row sizes, the number of members minus 2 must be a common multiple of 6 and 9.

Identify candidate numbers between 100 and 150 that are 2 more than a multiple of 18.

The candidate numbers are 110, 128, and 146.

To satisfy the condition of having a remainder of 2 when divided by 6 and 9, the total must be of the form

18k + 2

within the given range

[100, 150]

Check which candidate is divisible by 5.

110 is divisible by 5.

Since there are no members left over when lined up in rows of 5, the total number of members must be a multiple of 5.

Key Concept

Using the least common multiple (LCM) and remainders to solve divisibility word problems.

Estimated Time:1m 30s

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