Missing digits in a factorial

Here is a problem similar to a question which was on brilliant a few days ago:

Given that $37!=13763753091226345046315979581abcdefgh0000000$ , determine $a,b,c,d,e,f,g$ and $h$ .

Details and assumptions: It is expected that people refrain from using calculators and computers.

#NumberTheory #MathProblem #Math

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

We know that $37!$ contain no. $5$ occur $8-$ times and $2$ occur $34$ no. of times. So we can say that

$10$ occur $8$ times . bcz $10$ is a product of $5\times 2$ and there are $26$ no. of times $2$ remaining.

So we can say that $37!$ terminating with $8$ no. of zeros. So $h = 0$ .

Now for calculation of $a,b,c,d,e,f,g$ , we use divisibility test of $2$ .

Like $g$ must be divisible by $2$ and $f$ must be divisible by $2^2$ and and $e$ must be divisible by $2^3$

Similarly $d$ must be divisible by $2^4$ and and $c$ must be divisible by $2^5$ and $b$ must be divisible

by $2^6$ and $a$ must be divisible by $2^7$ .

jagdish singh - 7 years, 6 months ago

Can you elaborate a bit more on your technique of using divisibility tests of $2^{n}$ to find the rest of the digits. I can't quite clearly understand what you mean when you say " $e$ must be divisible by $2^{3}$ ... and so on". $e$ is a single digit number. So I can't comprehend your divisibility tests. I can sense what you are saying but probably you haven't expressed it in the right way. Correct me if I am wrong.

Bruce Wayne - 7 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$