Pretty obvious

Number Theory Level 1

1 ! 1! is divisible by 1

2 ! 2! is divisible by 2

3 ! 3! is divisible by 3

4 ! 4! is divisible by 4

5 ! 5! is divisible by 5

Is it true that 2019 ! 2019! is divisible by 2019?

Yes, it is true No, it is not true

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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3 solutions

Saksham Jain
Dec 4, 2017

we know x ! = x × ( x − 1 ) × ⋯ × 1 therefore x! is divisible by x \huge{x!=x\times(x-1)\times\dots\times1 \huge\text{therefore x! is divisible by x}}

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Annie Li
Dec 3, 2017

The reason why is because 1! contains 1

2! contains 2

3! contains 3 and so on so

2019! Must contain 2019

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Why you have stated (2019+2020+2021...) in your solution?

Munem Shahriar - 3 years, 6 months ago

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Oops 😬 soz, I will edit that

Annie Li - 3 years, 6 months ago
Matin Naseri
Feb 11, 2018

2 × 1 2 = 1 \frac{2×1}{2} = 1

3 × 2 × 1 3 = 2 × 1 \frac{3×2×1}{3} = 2×1

4 × 3 × 2 × 1 4 = 3 × 2 × 1 \frac{4×3×2×1}{4} = 3×2×1

5 × 4 × 3 × 2 × 1 5 = 4 × 3 × 2 × 1 \frac{5×4×3×2×1}{5} = 4×3×2×1

2019 × 2018 × . . . . × 2 × 1 2019 = 2018 × 2017 × . . . . × 2 × 1 \frac{2019×2018×....×2×1}{2019} = 2018×2017×....×2×1

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