Question

A marching band is struggling to line up for a parade. When they attempt to line up in groups of threes, twos are left out. When they try fours, three people remain unaligned. Similarly, when they attempt fives, four individuals are left out. What is the total number of people in the band?

A. 57
B. 67
C. 77
D. 87

Answer 1

The correct number of people in the marching band that fits the given conditions of leaving remainders 2, 3, and 4 when divided by 3, 4, and 5 respectively, is 87.

Step-by-step explanation:

The question involves finding the smallest number that has specific remainders when divided by 3, 4, and 5. This is a classic problem of number theory and can be solved using the Chinese Remainder Theorem or by systematic trial-and-error. To find the marching band's exact number, we look for a number 'n' such that:

• n % 3 = 2 (leaves a remainder of 2 when divided by 3)
• n % 4 = 3 (leaves a remainder of 3 when divided by 4)
• n % 5 = 4 (leaves a remainder of 4 when divided by 5)

To solve this, you can use the process of elimination with the given options:

• A. 57: 57 % 3 = 0, 57 % 4 = 1, 57 % 5 = 2
• B. 67: 67 % 3 = 1, 67 % 4 = 3, 67 % 5 = 2
• C. 77: 77 % 3 = 2, 77 % 4 = 1, 77 % 5 = 2
• D. 87: 87 % 3 = 2, 87 % 4 = 3, 87 % 5 = 2

The correct answer is D. 87, as it is the only option that fits all the given conditions.