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Number Theory Level 2

We all know that a b c ‾ ∣ 100 a + 10 b + 1 c \overline{abc} \ \Big| \ 100a + 10b + 1c because a b c ‾ = 100 a + 10 b + 1 c . \overline{abc} = 100a + 10b + 1c.

But are there any three distinct digits a , b , c a,b,c such that a b c ‾ ∣ 100 a 10 b 1 c ‾ ? \overline{abc} \ \Big| \ \overline{100a10b1c}\, ?

Yes, there are No, there aren't

Thành Đạt Lê by Thành Đạt Lê , 17, Singapore

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1 solution

Michael Mendrin
Jul 24, 2018

100110912 192 = 521411 \dfrac{100110912}{192}=521411 for example

Here are all the possible numbers

165 , 166 , 192 , 210 , 230 , 232 , 310 , 766 , 791 165, 166, 192, 210, 230, 232, 310, 766, 791

1 Helpful 0 Interesting 0 Brilliant 0 Confused

How many solutions are there?

Thành Đạt Lê - 2 years, 10 months ago

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with any digits = 0, or all a, b, c nonzero?

Michael Mendrin - 2 years, 10 months ago

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a a can't be zero. b b and c c can.

Thành Đạt Lê - 2 years, 10 months ago

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@Thành Đạt Lê – Just 9, then

Michael Mendrin - 2 years, 10 months ago

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@Michael Mendrin – Can you show all the solutions? (Sorry for taking your time away.)

Thành Đạt Lê - 2 years, 10 months ago

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