How many zeroes are there at the end of the following product? $1 \times 5 \times 10 \times 15 \times 20…
How many zeroes are there at the end of the following product?
$1 \times 5 \times 10 \times 15 \times 20 \times 25 \times 30 \times 35 \times 40 \times 45 \times 50 \times 55 \times 60$
10
12
14
15
Solution
The number of trailing zeroes equals the number of pairs of 2 and 5 in the prime factorization. The factors that are multiples of 5 are $5,10,15,20,25,30,35,40,45,50,55,60$ (twelve terms). Count powers of 5: each contributes at least one 5; $25$ and $50$ contribute an extra 5 each, so total power of 5 = $12 + 2 = 14$. Count powers of 2: from $10(1),20(2),30(1),40(3),50(1),60(2)$ = $1+2+1+3+1+2 = 10$. The number of trailing zeroes is the minimum of the two = $\min(14,10) = 10$.