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Question Number 1942 by Yozzi last updated on 25/Oct/15

Let N be a positive integer with prime factorisation N=p_1 ^m_1 p_2 ^m_2 p_3 ^m_3 ×...×p_(n−1) ^m_(n−1) p_n ^m_n where n,m_i ∈Z^+ and p_r is prime. How many proper factors does N have? Investigate cases where n=1,n=2, n=3 and n=4. What is the smallest positive integer with 12 proper factors? What is the smallest positive integer with at least 12 proper factors? (A proper factor of a positive number N is positive nteger M such that M≠1 and M≠N.)

$${Let}\:{N}\:{be}\:{a}\:{positive}\:{integer}\:{with}\:{prime} \\ $$$${factorisation}\: \\ $$$$\:\:{N}={p}_{\mathrm{1}} ^{{m}_{\mathrm{1}} } {p}_{\mathrm{2}} ^{{m}_{\mathrm{2}} } {p}_{\mathrm{3}} ^{{m}_{\mathrm{3}} } ×...×{p}_{{n}−\mathrm{1}} ^{{m}_{{n}−\mathrm{1}} } {p}_{{n}} ^{{m}_{{n}} } \\ $$$${where}\:\:{n},{m}_{{i}} \in\mathbb{Z}^{+} \:{and}\:{p}_{{r}} \:{is}\:{prime}. \\ $$$${How}\:{many}\:{proper}\:{factors}\:{does}\:{N}\:{have}? \\ $$$${Investigate}\:{cases}\:{where}\:{n}=\mathrm{1},{n}=\mathrm{2},\:{n}=\mathrm{3} \\ $$$${and}\:{n}=\mathrm{4}.\:{What}\:{is}\:{the}\:{smallest}\:{positive} \\ $$$${integer}\:{with}\:\mathrm{12}\:{proper}\:{factors}? \\ $$$${What}\:{is}\:{the}\:{smallest}\:{positive}\:{integer} \\ $$$${with}\:{at}\:{least}\:\mathrm{12}\:{proper}\:{factors}? \\ $$$$\left({A}\:{proper}\:{factor}\:{of}\:{a}\:{positive}\:{number}\:{N}\right. \\ $$$$\left.{is}\:{positive}\:{nteger}\:{M}\:{such}\:{that}\:{M}\neq\mathrm{1}\:{and}\:{M}\neq{N}.\right) \\ $$

Commented by prakash jain last updated on 25/Oct/15

Total factors,K=Π_(i=1) ^n (m_i +1) proper factors=K−2

$$\mathrm{Total}\:\mathrm{factors},\mathrm{K}=\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\left({m}_{{i}} +\mathrm{1}\right) \\ $$$$\mathrm{proper}\:\mathrm{factors}=\mathrm{K}−\mathrm{2} \\ $$

Answered by prakash jain last updated on 25/Oct/15

12 proper factor ⇒14 total factors 5×3=15>14 m_1 =4, m_2 =2 Smallest integer=2^4 ×3^2 =16×9=144

$$\mathrm{12}\:\mathrm{proper}\:\mathrm{factor}\:\Rightarrow\mathrm{14}\:\mathrm{total}\:\mathrm{factors} \\ $$$$\mathrm{5}×\mathrm{3}=\mathrm{15}>\mathrm{14} \\ $$$${m}_{\mathrm{1}} =\mathrm{4},\:{m}_{\mathrm{2}} =\mathrm{2} \\ $$$$\mathrm{Smallest}\:\mathrm{integer}=\mathrm{2}^{\mathrm{4}} ×\mathrm{3}^{\mathrm{2}} =\mathrm{16}×\mathrm{9}=\mathrm{144} \\ $$

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