But what's a humble number? (3)

Number Theory Level pending

A humble number is a number that consists of prime factors which are less than 10.


How many humble numbers does 210 210 divide?

210 ! 210! Infinitely Many 1 2 16 2^{16} 2 16 21 0 2 210^{2} 0

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1 solution

Efren Medallo
Jun 9, 2015

By definition, a humble number is a number whose prime factors consists of 2, 3, 5, or 7. It is a number of the form 2 a × 3 b × 5 c × 7 d 2^{a} \times 3^{b} \times 5^{c} \times 7^{d} , where a , b , c a, b, c , and d d are non-negative integers.

Since 210 210 is a humble number, it can divide any humble number of the form 2 a × 3 b × 5 c × 7 d 2^{a} \times 3^{b} \times 5^{c} \times 7^{d} , where a , b , c , a, b, c, and d d are integers all greater than or equal to 1.

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