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# number theory

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Find the smallest positive multiple of 21 that has no digit smaller than 5.

Apr 30, 2021

#1
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I don't know if there is some kind of algebraic way to find this (I'm pretty sure there isn't) so let's just list the multiples of 21.

21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 420, 441, 462, 483, 504, 525, 546, 567, 588, 609, 630, 651, 672, 693, 714, 735, 756, 777, 798, 819, 840, 861, 882, 903, 924, 945, 966, 987, 1008, 1029

In total, we have 7 positive multiples of 21 that have no digits smaller than 5. Now, we need to figure out the smallest one, which is easy:

The first green number is 567, which will therefore be our answer.

Ans = 567

Hope this helps :)

Apr 30, 2021
edited by NotGuest  Apr 30, 2021
#2
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I'm not quite sure if there's an algebraic way, but it might help to note that you don't have to check "21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 420, 441, 462, 483" since the hundreds digit will be smaller than 5. :))

=^._.^=

catmg  May 3, 2021