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Question Number 124567 by mr W last updated on 04/Dec/20

$$findthesmallestintegerwhichhas$$$$28divisorsandisdivisibleby28.$$

Answered by mr W last updated on 05/Dec/20

$$saythenumberisn=p^a{q}^b{r}^c...$$$$$$$$(a+1)(b+1)(c+1)...=28$$$$=28\times 1\Rightarrow a=27\Rightarrow n=p^{27}...\left(i\right)$$$$=14\times 2\Rightarrow a=13,b=1\Rightarrow n=p^{13}{q}^1...\left(ii\right)$$$$=7\times 4\Rightarrow a=6,b=3\Rightarrow n=p^6{q}^3...\left(iii\right)$$$$=7\times 2\times 2\Rightarrow a=6,b=1,c=1\Rightarrow n=p^6{q}^1{r}^1...\left(iv\right)$$$$$$$$28=2^2\times 7^1$$$$suchthatnisdivisibleby28,nmust$$$$containatleast{2}^{⩾2}{7}^{⩾1}$$$$case\left(i\right):$$$$impossible$$$$case\left(ii\right):$$$$n_{min}=2^{13}\times 7^1=57344$$$$case\left(iii\right):$$$$n_{min}=2^6\times 7^3=21952$$$$case\left(iv\right):$$$$n_{min}=2^6\times 3^1\times 7^1=1344$$$$\Rightarrow thenumberis1344.$$

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