# Real Numbers - HCF Word Problem

Ex 1.1 Q3. CBSE 10th Maths NCERT exercise solution

#### Question 3: 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?

Size of army contingent = 616 members
Size of band = 32 members

Condition: The two groups are to march in the same number of columns.
Let the number of columns be ‘n’.
Let the army contingent march in ‘a’ rows of ‘n’ columns.
Let the band march in ‘b’ rows of ‘n’ columns.
So, a × n = 616 and
b × n = 32
Number of columns and number of rows are integers.
‘n’ is, therefore, a factor of 616 and 32.

Objective: Compute maximum number of columns
i.e., find maximum or highest value of ‘n’ which is a common factor of 616 and 32.
i.e., Find the HCF of 616 and 32.

##### Compute HCF of 616 and 32

Step 1: Because 616 > 32, apply Euclid’s division lemma with 616 as the dividend and 32 as the divisor.
616 = 32 × 19 + 8
The remainder is not zero. So, recursively apply the Lemma as shown in step 2.

Step 2: Apply Euclid’s Division Lemma with 32 as dividend and 8 as divisor.
32 = 8 × 4 + 0
The remainder is zero at Step 2.
∴ HCF of 616 and 32 is the divisor of this step. The HCF is 8.
The maximum number of columns is 8.

If they are arranged in 8 columns to a row, number of rows = $\frac{648}{8}$ = 81.