George’s new office has a security
lock in which you have to key in a series of digits, all different, before you
can open the door.
Unfortunately, he
is having trouble committing this security number to memory. So far, he
has memorized a number consisting
a selection of different digits from the
security number.
He made a note of
this memorized number on the memo
sheet on which he was given the security number, but he absent-mindedly
left the paper lying around the house.
When his wife found the paper containing the two numbers, she multiplied them together and found the product
was a seven-digit number in which all
the digits were the same.
What is
George’s security number?
The prime factorization of 1111111 = 239*4649
We need a*239 and b*4649 each to have unique digits and all the digits of a*239 are found in b*4649, and also a*b<10
The multiples of b*4649 with unique digits:
3*4649=13947
4*4649=18596
6*4649=27894
8*4648=37192 <--- note, this contains 239 so it's a solution
Any other values of a?
2*239=478, neither b=3 or b=4 contains these
3*239=717
So the only solution is a=1, b=8.
239 contains the digits of 37192. The product of these is 8888888.
[I made a complete (a,b) list up to 10. There are two non-solutions (2,6) and (4,4) which give products of 1111111*12 and 1111111*16.]
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Posted by Jer
on 2024-06-13 17:38:28 |
Calculator solution
