# Quickie #5

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

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

Identify the digits.
• Dec 29th 2006, 02:31 PM
OReilly
Quote:

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

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

Identify the digits.

$7(461538)=6(538461)$
• Dec 29th 2006, 03:48 PM
Soroban
Hello, OReilly!

Your solution is correct.

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

• Dec 29th 2006, 06:09 PM
galactus
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:

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

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

$6994f=5993f$

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

$538f=461h$

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

$7(461538)=6(538461)$
• Dec 30th 2006, 01:08 AM
earboth
Quote:

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

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

Identify the digits.

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 AM
Soroban
Lovely, Galactus!

That is exactly the Quickie solution.

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

Since $638$ and $461$ are relatively prime (you already divided out the GCD),
. . the only solution in positive integers is: . $f = 461,\:h = 538$

• Dec 30th 2006, 05:18 AM
galactus
Quote:

Originally Posted by Soroban
Lovely, Galactus!

That is exactly the Quickie solution.

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

Since $638$ and $461$ are relatively prime (you already divided out the GCD),
. . the only solution in positive integers is: . $f = 461,\:h = 538$

Even a blind hog finds an acorn once in a while.