In the following cryptarithm, each letter represents a distinct digit (base 10):

. . . . . $\displaystyle 7(FRYHAM) \:=\:6(HAMFRY)$

Identify the digits.

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- Dec 29th 2006, 06:34 AMSorobanQuickie #5
In the following cryptarithm, each letter represents a distinct digit (base 10):

. . . . . $\displaystyle 7(FRYHAM) \:=\:6(HAMFRY)$

Identify the digits. - Dec 29th 2006, 02:31 PMOReilly
- Dec 29th 2006, 03:48 PMSoroban
Hello, OReilly!

Your solution is correct.

You offer no explanation . . . did you use some software?

- Dec 29th 2006, 06:09 PMgalactus
Hey Soroban:

I scratched my head on this one for a while.

One could tackle this in many ways, but if we let FRY=f and HAM=h, then

we can whittle it down to something easier to solve.

Then we have:

$\displaystyle 7(1000f+h)=6(1000h+f)$

$\displaystyle 7000f+7h=6000h+6f$

$\displaystyle 6994f=5993f$

$\displaystyle gcd(6994,5993)=13$. Divide through by 13.

$\displaystyle 538f=461h$

Now, the ol' switcharoo.

I was amazed this actually worked.

I didn't see the very last part right away.

Because we can switch them around and have 538461 and 461538

$\displaystyle 7(461538)=6(538461)$ - Dec 30th 2006, 01:08 AMearboth
Hello Soroban,

your problem needs "nothing else" but counting. So I wrote a small program (see attachment) and the result popped up rather immediately. There isn't any mathematical thinking or deduction involved so I didn't dare to publish my "solution". so this post is only for the records.

EB - Dec 30th 2006, 02:21 AMSoroban
Lovely, Galactus!

That is*exactly*the Quickie solution.

Punchline: .Your final equation was: .$\displaystyle 538f = 461h$

Since $\displaystyle 638$ and $\displaystyle 461$ are relatively prime (you already divided out the GCD),

. . the only solution in positive integers is: .$\displaystyle f = 461,\:h = 538$

- Dec 30th 2006, 05:18 AMgalactus