In the following cryptarithm, each letter represents a distinct digit (base 10):
. . . . . $\displaystyle 7(FRYHAM) \:=\:6(HAMFRY)$
Identify the digits.
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
$\displaystyle 7(461538)=6(538461)$
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
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: .$\displaystyle f = 461,\:h = 538$