# Quickie #5

• Dec 29th 2006, 06:34 AM
Soroban
Quickie #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 PM
OReilly
Quote:

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

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

Identify the digits.

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

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:

\$\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 AM
earboth
Quote:

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

. . . . . \$\displaystyle 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: .\$\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 AM
galactus
Quote:

Originally Posted by Soroban
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\$

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