Does it?

Number Theory Level 2

Does there exist a number that starts with 9876543210 and at the same time it is divisible by 2017?

of course it does! no way...

1 solution

Zee Ell
Apr 15, 2017

There are 10,000 consecutive integers between 98,765,432,100,000 and 98,765,432,109,999, all starting with 9876543210.

By either considering simple facts regarding division or by applying the pigeonhole principle, we can find, that there are at least 4 suitable numbers within this interval:

$\lfloor 10000 ÷ 2017 \rfloor = 4$ .

$\text {Hence, our answer should be: } \boxed { \text {of course it does! } }$

Does it?

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