A can X contains 399 litres of petrol and a can Y contains 532 litres of diesel. They are to be bottled in bottles of equal size so that whole of petrol and diesel would be separately bottled. The bottle capacity in terms of litres is an integer. How many different bottle sizes are possible?

(a) 3
(b) 4 ✓ UPSC's answer
(c) 5
(d) 6

Why the answer is (b)

• The problem requires finding the number of integer bottle sizes that can exactly divide both 399 litres of petrol and 532 litres of diesel.

• This is equivalent to finding the number of common divisors of 399 and 532, which is the number of divisors of their Greatest Common Divisor (GCD).

• Prime factorization of 399 is 3 × 7 × 19.

• Prime factorization of 532 is 2² × 7 × 19.

• The GCD of 399 and 532 is 7 × 19 = 133.

• The divisors of 133 are 1, 7, 19, and 133, totaling 4 possible bottle sizes.

Why the other options are wrong

(a) 3: Option (a) is incorrect because there are 4 common divisors, not 3.
(c) 5: Option (c) is incorrect because 133 has only 4 divisors, not 5.
(d) 6: Option (d) is incorrect because the GCD is 133, which has 4 divisors, not 6.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.