A number has 24 factors.The multiplication of its prime factors is 30(taking 1 at a time).What is the ratio of the square roots of the highest and the smallest such number possible?
The hard part is the first sentence. You know the primes that divide such a number (as these are the primes that multiply to get 30), and the powers of the primes sum to 24 and are all non-zero (as otherwise the product would not be 24). So, what is the largest number possible? What is the smallest?
Big hint: let $\displaystyle x=p_1^{a_1}p_2^{a_2} \ldots p_n^{a_n}$ be such a number, with $\displaystyle p_1<p_2 < \ldots < p_n$. Then the largest is the one with $\displaystyle a_1=a_2= \ldots = a_{n-1} = 1$ and $\displaystyle a_n = 24-(n-1)$.
(Unless I'm missing something here?)
Hello,
For example, 6 has 4 factors : 1,2,3,6
If I'm not mistaking (hasn't seen this for a while) : the number of factors of a number $\displaystyle n=p_1^{\alpha_1}\cdot p_2^{\alpha_2} \cdots p_m^{\alpha_m}$ is $\displaystyle \prod_{k=1}^m \{\alpha_k+1\}$
So here, we have $\displaystyle \prod_{k=1}^m \{\alpha_k+1\}=24$
And the product of its prime factors is 30 : $\displaystyle \prod_{k=1}^m p_k=30$
Then, we have to be able to find the smallest and greatest possible numbers responding to these conditions... So I'm pretty sure we have to study the prime decomposition of 30
But now... I have to take a shower
All right guys, I just remembered this: the tau function $\displaystyle \tau(n)$ which gives the number of factors of each integer $\displaystyle n > 1.$ $\displaystyle \tau$ has the following property: if $\displaystyle n=p_1^{k_1}\cdots p_r^{k_r}$ where the $\displaystyle p_i$ are distinct primes, then
$\displaystyle \tau(n)\ =\ \prod_{i\,=\,1}^r(1+k_i)$
So for this particular problem, the distinct primes are 2, 3, and 5 and the number is of the form $\displaystyle 2^a3^b5^c$ where $\displaystyle (1+a)(1+b)(1+c)=24.$ The possibilities are thus
(i) $\displaystyle a=1,\,b=1,\,c=5$
(ii) $\displaystyle a=1,\,b=5,\,c=1$
(iii) $\displaystyle a=5,\,b=1,\,c=1$
(iv) $\displaystyle a=1,\,b=2,\,c=3$
(v) $\displaystyle a=1,\,b=3,\,c=2$
(vi) $\displaystyle a=2,\,b=1,\,c=3$
(vii) $\displaystyle a=2,\,b=3,\,c=1$
(viii) $\displaystyle a=3,\,b=1,\,c=2$
(ix) $\displaystyle a=3,\,b=2,\,c=1$
The largest number is therefore $\displaystyle 2^13^15^5$ and the smallest is $\displaystyle 2^33^25^1.$ Hence the answer to the question is $\displaystyle \sqrt{\frac{2^13^15^5}{2^33^25^1}}\,=\,\frac{25}{2 \sqrt3}.$