SOLUTION: I'm thinking of an interesting number that I'll call N. When N has a 7 in front of it, the number formed is 5 times as large as when N has a 7 after it. The smallest number of digi

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Question 1061453: I'm thinking of an interesting number that I'll call N. When N has a 7 in front of it, the number formed is 5 times as large as when N has a 7 after it. The smallest number of digits in N is:
a) 3 b)4 c)5 d)6 e)7
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
d= number of digits N has.
7%2A10%5Ed%2BN = N with a 7 added as a first digit
10N%2B7 = N with a 7 added as a last digit
The problem says that
N%2B7%2A10%5Ed=5%2810N%2B7%29
N%2B7%2A10%5Ed=50N%2B35
7%2A10%5Ed-35=50N-N
7%2810%5Ed-5%29=49N
10%5Ed-5=7N
You could see what value of d makes 10%5Ed-5
a multiple of 7 ,
but I want easier calculations
,
so I am looking for how to make a multiple of 7 out of
10%5Ed-5=7K%2B3%5Ed-5 ,
which means 3%5Ed-5 being a multiple of 7.
d%3C=4 does not work:
3%5E2-5=9-5=4
3%5E3-5=27-5=22=21%2B1=7%2A3%2B1
3%5E4-5=81-5=76=70%2B6=7%2A10%2B6
For highlight%28d=5%29 ,
3%5E5-5=243-5=238=7%2A34 ,
we find the first multiple of 7.