How many zeroes are there at the end of the following product? 1 × 5 × 10 × 15 × 20 × 25 × 30 × 35 × 40 × 45 × 50 × 55 × 60

(a) 10 ✓ UPSC's answer
(b) 12
(c) 14
(d) 15

Why the answer is (a)

• The number of trailing zeroes in a product is determined by the number of pairs of prime factors 2 and 5, where the count of 5s is typically the limiting factor.

• The product consists of multiples of 5 from 5 to 60: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60.

• Count the factor 5 in each term: 5 (1), 10 (1), 15 (1), 20 (1), 25 (2), 30 (1), 35 (1), 40 (1), 45 (1), 50 (2), 55 (1), 60 (1).

• Summing these gives a total of 14 factors of 5.

• Count the factor 2 in each term: 10 (1), 20 (2), 30 (1), 40 (3), 50 (1), 60 (2). The term 5, 15, 25, 35, 45, 55 have no factor 2. Total factors of 2 = 1+2+1+3+1+2 = 10.

• Since there are only 10 factors of 2, they form 10 pairs with the available 5s, resulting in 10 trailing zeroes.

Why the other options are wrong

(b) 12: Option (b) is incorrect because it likely counts the total number of factors of 5 (which is 14) or makes an arithmetic error in counting factors of 2, ignoring that factors of 2 are the limiting reagent.
(c) 14: Option (c) is incorrect because it equals the total count of factors of 5 in the product, failing to account for the lower count of factors of 2 which limits the number of 10s.
(d) 15: Option (d) is incorrect because it overestimates the number of available factors of 2 or incorrectly assumes an equal distribution of 2s and 5s.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.