Question

The director of a marching band asks the band members to line up in rows of four, but one is left over. Then she tries to line them up in rows of six, but three are left over. Finally, she tries to line them up in rows of seven, but four are left over. The band has fewer than 100 members. How many members are there?

Answer 1

There are 81 band members. :-)

4 rows of 20 = 80 ... with one left over = 80 + 1 (81)
6 rows of 13 = 78 ... with three left over = 78 + 3 (81)
7 rows of 11 = 77 ... with four left over = 77 + 4 (81)

Answer 2

The answer is 81.

Explanation:

When the band members are told to line up in rows of four, but one is left over. This means the overall number id an odd number.

When they are lined up in rows of six, but three are left over. So, eliminate numbers with 3, 5, 6 ,9, 0.

Finally they are lined up in rows of seven, but four are left over.

So, eliminate numbers with 7 also.

So, all eliminated numbers are : 2,3,4,5,6,7,8,9,10

Left numbers are 1.

Given is that the band has fewer than 100 members. So, it can be 71 ,81 ,91

We can check with 81.

81/4 = 20 remainder 1

81/6 = 13 remainder 3

81/7 = 11 remainder 4

So, the answer is 81.