Solution:
Given that HCF of 144 and 180 is 13m – 16.We know that, Euclid's Division Lemma is stated as “For any two positive integers say a and b there exist two unique whole numbers say q and r, such that, a = bq + r, where 0 ≤ r <b.
Let us apply Euclid’s division Lemma for 180 and 144.
Here, 180 > 144.
Let a = 180 and b = 144.
So, 180 can be written in the form of a = bq + r as,
180 = 144 ×1 + 36
Here, Remainder = 36 ≠ 0
So, consider a new Dividend as 144 and divisor as 36.
Now again apply, Euclid’s division Lemma for 144and 36.
Here, 144 > 36
Let a = 144 and b = 36.
So, 144 can be written in the form of a = bq + r as,
144 = 36 × 4 + 0
Here, the remainder is zero and the divisor is 36.
Hence, the HCF of 144 and 180 is 36.
But it is given HCF of 144 and 180 is 13m – 16.
So,
13m - 16 = 36
13m = 36 + 16
13m = 52
m = 52/13
m = 4
Hence, the value of m is 4.
The correct option is (4).
