Finding Highest Power of Number

Concepts Examples

Finding highest power of a number can be used to find the

1. Highest power of a number that divide the factorial
2. Highest power of a product of numbers that divide the factorial
3. Number of 0's at the end of the factorial

To find the highest power of a number 'x' in the factorial of number N,

Step 1 : Divide the number N successively till we get 0 as the last quotient.

Step 2 : Add all the quotients to get the highest power that divides the factorial of N

e.g Find the highest power of a 2 in factorial 10.

i) To find highest power of 2 in 10! we divide 10 successively

2	10
2	5
2	2
2	1
	0

ii) Add all the quotients, i.e. 5+2+1=8

∴ HIghest power of 2 in 10! is 8

Question 1 What will be remainder when 888³³³+222⁸⁸⁸ is divided by 5

1	2
3	4

A

Remainder of

$\frac{888^{222}+2^{888}}{5}$888222+28885 

= Remainder of

$\frac{3^{222}}{5}+\frac{2^{888}}{5}$32225 +28885 

= Remainder of

$\frac{9^{111}}{5}+\frac{4^{444}}{5}$91115 +44445 

= Remainder of

$\frac{-1^{111}}{5}+\frac{-1^{444}}{5}$-11115 +-14445 

= Remainder of

$\frac{-1+1}{5}$-1+15 

= 0

Thus 888³³³+222⁸⁸⁸ is divisible by 5 and leaves 0 as the remainder

Question 2 Find the highest power of 3 that can divide 333!

165	164
191	111

A

3	333
3	111
3	37
3	12
3	4
3	1
3	0

So highest power of 3 in factorial 333 = 111+37+12+4+1 = 165

Question 3 Find the highest power of 10 that can divide 55!

50	25
13	12

C

10 = 2¹ × 5¹

2 55 2 27 2 13 2 6 2 3 2 1

5 55 5 11 5 2 5 0

So highest power of 2 in factorial 55 = 27 + 13 + 6 + 3 + 1 = 50

So highest power of 5 in factorial 55 = 11 + 2 = 13

∴ HIghest power of 10 in factorial 55 = 13 (since highest power of 5 is only 13 while power of 2 is 50, thus power of 5 is the limiting factor here)

Question 4 Find the highest power of 2 in 879!

879	871
850	820

B

2	879
2	439
2	219
2	109
2	54
2	27
2	13
2	6
2	3
2	1
2	0

So highest power of 2 in factorial 879! = 439 + 219 + 109 + 54 + 27 + 13 + 6 + 3 + 1 = 871

Question 5 Find the highest power of 20 that can divide 200!

197	49
98	150

B

20 = 2² × 5¹

2 200 2 100 2 50 2 25 2 12 2 6 2 3 2 1 2 0

5 200 5 40 5 8 5 1

So highest power of 2 in factorial 200 = 100+50+25+12+6+3+1 = 197

So highest power of 5 in factorial 200 = 40+8+1 =49

∴ Highest power of 4 in factorial 200 = 197/2 = 98

∴ HIghest power of 20 in factorial 200 = 49 (since highest power of 5 is only 49 while power of 4 is 98, thus power of 5 is the limiting factor here)

Question 6 Find the highest power of 5 in 125!

25	27
31	119

C

5	125
5	25
5	5
5	1
5	0

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