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How to prove that $$\frac{1000!}{(500!)^2}$$ is not divisible by 7?

I reduced the above fraction to: $$\frac{\prod_{k=501}^{1000}k}{500!} $$

But I don't know how to proceed. Please help.

  • 0
    Hint: consider prime factorisations. Focus on $7$'s. – 2017-01-24

5 Answers 5

2

In $\{1,\dots, 500\}$, there are 71 multiples of 7, 10 multiples of $7^2$ and one multiple of $7^3$.
How many multiples of $7$, $7^2$, $7^3$ do you have in $\{501,\dots, 1000\}$? Same amount.

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    Why are they the same amount? – 2017-01-24
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    504 is first multiple of 7 (with multiplicity of seven is 1 in factorization), and $504 + 7 \cdot 70 = 994$. Also $7^2 \cdot 11 = 539$ (multiplicity is 2) and $7^2 \cdot 20 = 980$. At last $7^3 \cdot 2 = 686$ (fits only once). – 2017-01-24
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This is overkill, but by Lucas's Theorem$$\frac{1000!}{(500!)^2}=\binom{1000}{500} \equiv \binom{2}{1}\binom{6}{3}\binom{2}{1}\binom{6}{3} \equiv 4 \pmod{7}$$ As $1000=2626_{7}$ and $500=1313_{7}$.

Here is another , more elementary argument, from Legendre's formula, $$\nu_7(n!) = \frac{n - s_7(n)}{6}$$ So $\nu_7(100!)=164, \nu_7 (50!)=82$. So the $7$ s in the denominator and the numerator cancel out. So we have the desired result.

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    Got it! But you told that applying Lucas's Theorem is overkill, so, is there any other solution? – 2017-01-24
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    @AmitayasBanerjee I added one right now. This one is a more elementary result. – 2017-01-24
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    (+1) Two beautiful proofs in a single answer make a well-deserved upvote. – 2017-01-24
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    [+1] I love Lucas' theorem, and a good application of it. – 2017-01-24
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    @AmitayasBanerjee Could you tell me why you unaccepted my answer? – 2017-04-29
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It is a general formula that the number of times a prime $p$ occurs in the prime factorisation of $n!$ is given by

$$ \left[\frac{n}{p}\right] + \left[\frac{n}{p^2}\right] + \left[\frac{n}{p^3}\right] + \cdots $$

where $[x]$ denotes the greatest integer $≤x$.

In $1000!$ you'd have

$$ \left[\frac{1000}{7}\right] + \left[\frac{1000}{7^2}\right] + \left[\frac{1000}{7^3}\right] + \cdots = 164 $$

$7$'s, and in $500!$ you have

$$ \left[\frac{500}{7}\right] + \left[\frac{500}{7^2}\right] + \left[\frac{500}{7^3}\right] + \cdots = 82 $$

$7$'s. So you'll have a total of $164$ 7's in $(500!)^2$.

The rest follows.

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    I am sorry but would you kindly help in the conclusion? I scratched my head for a while but did not understand how that proves that it is not divisible by 7. – 2018-11-14
5

You can also overkill it by Kummer's theorem:

Notice that $500$ is $1313$ in base $7$. Since every digit is smaller than $\frac{7}{2}$ there are no carries when adding $1313+1313$ in base $7$.

Therefore $\binom{1000}{500}$ is not a multiple of $7$.

In general we have that a prime $p$ divides $\binom{2n}{n}$ if and only if $n$ has at least one digit larger than or equal to $p/2$ in its base $p$ expansion.

4

Consider prime factorisations:

$$\lfloor\frac{1000}{7}\rfloor+\lfloor\frac{1000}{7^2}\rfloor+\lfloor\frac{1000}{7^3}\rfloor+...=142+20+2=164$$ Then $$7^{164}|1000!$$

Similarly:

$$\lfloor\frac{500}{7}\rfloor+\lfloor\frac{500}{7^2}\rfloor+\lfloor\frac{500}{7^3}\rfloor+...=71+10+1=82$$ Then $$7^{164}=7^{82\cdot2}=(7^{82})^2|(500!)^2$$

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    Sorry, but didn't get it. – 2017-01-24
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    The fact that $7^{164}$ divides $1000!$ and $(500!)^2$ does not suffice to decide if $7$ divides their ratio or not. Please be more careful. – 2017-01-24
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    Yes. But I considered prime factorisations – 2017-01-24
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    So the point is not that $7^{164}$ divides $1000!$, but that $7^{165}$ does not. – 2017-01-24
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    I'm understood, thank you! – 2017-01-24