An army contingent of $616$ members is to march behind an army band of $32$ members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

HCF ($616,32$) is the maximum number of columns in which they can march.

Step 1: First find which integer is larger.

$616>32$

Step 2: Then apply the Euclid's division algorithm to $616$ and $32$ to obtain

$616=32\times 19+8$

Repeat the above step until you will get remainder as zero.

Step 3: Now consider the divisor 32 and the remainder 8, and apply the division lemma to get

$32=8\times 4+0$

Since the remainder is zero, we cannot proceed further.

Step 4: Hence the divisor at the last process is $8$

So, the H.C.F. of $616$ and $32$ is $8.$

Therefore, $8$ is the maximum number of columns in which they can march.