Divisible by 3? (3)

Number Theory Level 2

How many of the following are divisible by 3?

1 ! , 2 ! , 3 ! ⋯ 999 ! , 1000 ! \large 1!, 2! , 3! \cdots 999! , 1000!

Notation: ! ! denotes the factorial notation. For example: 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times 2 \times 3 \times \cdots \times 8 .


The answer is 998.

A W by A W , 17, United Kingdom

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2 solutions

Rajdeep Ghosh
May 23, 2017

Too easy. 1 ! = 1 1!=1 2 ! = 1 ∗ 2 2!=1*2 3 ! = 1 ∗ 2 ∗ 3 3!=1*2*3 4 ! = 1 ∗ 2 ∗ 3 ∗ 4 4!=1*2*3*4 After 2! every factorial contains 3. So answer is 1000-2=998

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Sumukh Bansal
Nov 19, 2017

This is a very easy problem. 1 ! = 1 1!=1 2 ! = 1 × 2 2!=1\times2 3 ! = 1 × 2 × 3 3!=1\times2\times3 1000 ! = 1 × 2 × 3 ⋯ 1000 1000!=1\times2\times3\cdots1000 As in all the factorials 3 is multiplied except in 1! and 2! so answer is 1000-2=998

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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