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?

Hint – Here we will proceed from the integer which is larger and then apply Euclid’s division lemma to both the integers. Then we will repeat the algorithm up to the time we get remainder as zero. Hence we will get the desired result.

Complete step-by-step answer:
According to Euclid’s division lemma, if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition $a = b \times q + r$ where $0 < = r < = b$.
Firstly, we will find which integer is larger.
$ \Rightarrow 616 > 32$
Then we will apply the Euclid’s division lemma to 616 and 32 to obtain-
$ \Rightarrow 616 = 32 \times 19 + 8$
We will repeat the above step until we get remainder as zero.
$ \Rightarrow 32 = 8 \times 4 + 0$
Since we got the remainder, we cannot proceed further.
Hence the divisor at the last process is 8.
So, the HCF of 616 and 32 is 8.
Therefore, 8 is the maximum number of columns in which they can march.
Note- In this type of question, we must understand that for calculating maximum, we use HCF and for calculating minimum, we use LCM. Also one may omit the first step mentioned above i.e. finding the large integer which will not give the right answer.