Variation of a previous number theory problem

September 19th, 2009 by ben Leave a reply »

Lol, just saw this on the math team forum, was posted by like last last year, but anyways…

In the prime factorization of the expression 100! \cdot 99! \cdot 98! \cdot \ldots \cdot 3! \cdot 2! \cdot 1! there is a 7^x term. Find x.

So, I’m gonna start out by figuring out how many 7s are in 100! .

\dfrac{100}{7} \to \dfrac{14}{7} \to 2

Therefore there are 14 + 2 factors of 7 in 100! . However, it’s going to take a while to figure out everything down all the way to 1! .

There’s probably a smarter way of doing this, since I’m just dumb, but here is how my feeble mind tackles it.

Smallest 7 is in 7! , occurs 100 - 7 + 1 = 94 times.

Largest 7 is 98. This occurs three times, in 98, 99, and 100 factorial.

Therefore, we have the summation:

\displaystyle\sum^{14}_{i = 1} 101 - 7i

We also need to count 7^2 = 49 .

Smallest 49 is in 49! , occurs 100 - 49 + 1 = 52 times.

The only other 49 is in 98! until 100, so it occurs 100 - 98 + 1 = 3 times.

The solution should be:

\left( \displaystyle\sum^{14}_{i = 1} 101 - 7i \right) + \left( \displaystyle\sum^{2}_{i = 1} 101 - 49i \right)

And since we know the second sum already (as it only has two terms anyways and we’ve calculated them to be 52 + 3 times already), we can replace them.

\left( \displaystyle\sum^{14}_{i = 1} 101 - 7i \right) + \left( 55 \right)

Use arithmetic sequence formula or whatever to solve the summation.

\left( \dfrac{14(3 + 94)}{2} \right) + \left( 55 \right)

My answer is:

( 7 \cdot 97 ) + 55 = 679 + 55 = \boxed{734}
Proof that there are indeed 734 factors of 7 in the number <img src='http://s0.wp.com/latex.php?latex=100%21+%5Ccdot+99%21+%5Ccdot+%5Cldots+%5Ccdot+2%21+%5Ccdot+1%21+&bg=ffffff&fg=000000&s=0' alt='100! \cdot 99! \cdot \ldots \cdot 2! \cdot 1! ' title='100! \cdot 99! \cdot \ldots \cdot 2! \cdot 1! ' class='latex' />.

Proof that there are indeed 734 factors of 7 in the number 100! * 99! * ... * 2! * 1!.

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2 comments

  1. k says:

    “Math team forum” sounds incredibly nerdy.

  2. Benji says:

    It’s not the nerdiest forum out there, you know.

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