Finding Number of Zeros in a Number

Concepts Examples

To find the number of zeros at the end of a factorial or a product, find the highest power of 2 and 5 in the factorial or the product, number of zeros will be decided by the limiting factor. ( one who's power is smaller will be the limiting factor)

Limiting factor is the one who's power is less among 2 and 5

e.g. Finding number of Zeros in 150

150 = 2 × 3 × 5 × 5,
∴ Highest power of 2 = 1
∴ Highest power of 5 = 2
∴ Limiting factor = 1
∴ Number of zeros in 150 = 1

e.g. Finding number of Zeros in 10!

∴ Highest power of 2 = 8
∴ Highest power of 5 = 2
∴ Limiting factor = 2
∴ Number of zeros in 10! = 2

Question 1 Find the number of zeros at the end of 33 × 128 × 125 × 50 × 45

3	4
5	6

D

33 × 128 × 125 × 50 × 45 = 2⁸ × 3³ × 5 ⁶ × 11¹

∴ Number of zeros = 6

Question 2 Find the number of zeros at the end of 100!

97	56
24	23

C

10 = 2¹ × 5¹

2 100 2 50 2 25 2 12 2 6 2 3 2 1

5 100 5 20 5 4

So highest power of 2 in factorial 100 = 50+25+12+6+3+1 = 97

So highest power of 5 in factorial 100 = 20+4 = 24

So limiting factor is 5 and the number of zeros at end of 100! = 24

Question 3 Find the number of zeros at the end of 125!

119	31
27	29

B

10 = 2¹ × 5¹

2 125 2 62 2 31 2 15 2 7 2 3 2 1

5 125 5 25 5 5 5 1

So highest power of 2 in factorial 125 = 62+31+15+7+3+1 = 119

So highest power of 5 in factorial 125 = 25+5+1 = 31

So limiting factor is 5 and the number of zeros at end of 125! = 31

Question 4 Find the number of zeros at the end of 390625 × 1024

8	10
15	17

A

390625 = 5⁸

1024 = 2¹⁰

Thus the limiting factor will be 5 here and the highest power of 10 will be 8

Thus the number of zeros at the end of the 390625 × 1024 = 8