Question

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?

Answer 1

Hint:
This question is based on the Euclid’s division lemma. Here we will proceed with the larger integer and then we apply Euclid’s division lemma to both integers. We will repeat the same procedure until we get the remainder as zero.

Complete step by step solution:
Here to solve this question we will use Euclid’s division lemma.
According to Euclid division lemma for two positive integers \[a\] and \[b\] there exists unique integers \[q\] and \[r\] such that \[a = b \times q + r\] where \[0 \le r \le b\].
Now we will first find which integer is larger.
\[616 > 32\]
Applying the Euclid’s division lemma to 616 and 32, we get
\[616 = 32 \times 19 + 8\]
We will now repeat the above step until we get remainder as zero.
Now considering the divisor 32 and the remainder 8 and applying the Euclid’s division lemma, we get
32 = 8 × 4 + 0 \[32 = 8 \times 4 + 0\]
Since here we get the remainder as zero, we cannot proceed further.
As the divisor at the last step 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:
To solve this question, we need to check whether we have to calculate the maximum number or minimum number. In the case of the maximum number, we have to find the HCF of those numbers and in the case of a minimum number, we have to find the LCM of those numbers. Euclid’s division lemma is used to find the HCF (Highest Common Factor) of two positive integers.