Concepts
Examples

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

- Highest power of a number that divide the factorial
- Highest power of a product of numbers that divide the factorial
- 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

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 888^{333}+222^{888} is divided by 5

A

Remainder of

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

= Remainder of

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

= Remainder of

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

= Remainder of

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

= Remainder of

$\frac{-1+1}{5}$-1+15

= 0

Thus 888

^{333}+222

^{888} is divisible by 5 and leaves 0 as the remainder

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

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!

C

10 = 2

^{1} × 5

^{1} 2 | 55 | 2 | 27 | 2 | 13 | 2 | 6 | 2 | 3 | 2 | 1 | | |

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!

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!

B

20 = 2

^{2} × 5

^{1} 2 | 200 | 2 | 100 | 2 | 50 | 2 | 25 | 2 | 12 | 2 | 6 | 2 | 3 | 2 | 1 | | 2 | 0 | | |

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!

C

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

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