# The prime numbers that divide $10^4-1$

How to find the prime numbers that divide $10^4-1$

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What have you tried? –  Tobias Kildetoft Feb 11 '13 at 14:27

HINT

1. $x^2 - y^2 = (x+y)(x-y)$
2. $101$ is prime
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Hint: $10^4 - 1 = (100 + 1)(100 - 1) = 3^2 \times 11 \times 101$

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No. $101=99+2$ not divisible by $3$, neither by $5$ or $7$ or $11$, so, it is prime. –  Berci Feb 11 '13 at 14:39
But $3\times 37=111$, not $101$. –  Per Manne Feb 11 '13 at 14:40
@PerManne Edited :-) –  Parth Kohli Feb 11 '13 at 14:40

$$10^4-1=(10^2-1)(10^2+1)=(10-1)(10+1)(101)=9\times11\times101$$ which implies that the prime divisors are $3,11$ and $101$.

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$$9999:9=1111 \\ 1111:11=101$$ So, $9999=3^2\cdot 11\cdot 101$.

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I have a question/comment about readability: why use colons rather than slashes? –  rschwieb Feb 11 '13 at 15:02
@rschwieb, it's the common division notation in some locales... –  vonbrand Feb 11 '13 at 18:41
@vonbrand I don't have any on hand, but I would venture to guess that far more people would recognize $9999/9=1111$. It's not a big deal since the context is so simple, but I think it's a fact to bear in mind for future posts. –  rschwieb Feb 11 '13 at 21:29