Math Is Fun Forum

mrpace

Member

Registered: 2012-08-16

Posts: 88

Find the smallest prime factor of 10^101 + 1

I think it's fairly intuitive that the answer is 11 here, but I'm wondering if there is a more concrete/elegant way of solving this?

Thanks

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Find the smallest prime factor of 10^101 + 1

hi mrpace

You can eliminate the earlier prime factors like this

2?  The number is not even so 2 isn't a factor.

3? If you take any number in the three times table and add its digits you get another (smaller) number that is in the three times table.  eg. 48 4+8 = 12

As the digits add to 2 we can eliminate 3 as a factor.

5? Number has to end in 5 or 0, so 5 isn't a factor.

7?  This is the hardest so far to test.  If you divide by 7 you get

142857152857142857.....14285 remainder 6 so 7 isn't a factor.

As you say 11 does work ( add alternate sets of digits to get a difference of 0) so it's the smallest.

Bob

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Children are not defined by school ...........The Fonz
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Sometimes I deliberately make mistakes, just to test you!  …………….Bob