A number has exactly 8 factors . Two of the factors are 27 and 75. List all the factors of the given Number .
I find LCM of 27 and 75 and the LCM value is the Number. I did this luckily and I got correct but I do not understand the real reason why the 'number' is LCM of 27 and 75 . Maybe it's because of my concept problems ..
On
Well, I hope you realize why it has to be a common multiple of the two factors. So the least common multiple will be a factor.
If $27=3^3$ is a factor than so are all its factors: 1,3,9,27. If $75=3*5^2$ is a factor then so are all its factors: 1, 3, 5, 25, 15, 75.
Combining the two you have the possible factors: $1,3,3^2=9,3^3 =27, 5,5*3=15,5*9=45,5*27 = 135, 25,25*3=75,25*9=225, 25*27=675$. That is 12 possible factors which ... is not what the problem stated. Even if I remove 1, and "the number" 675, we have 10 proper factors.
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Another way of looking at is is that if a number is $N= \prod_i p_i^{k_i}$ where $p_i$ are the prime factors then $N$ will have $\prod (k_i + 1)$ factors. Example $675 = 27*25 = 3^3*5^2$ will have $(3+1)(2+1) = 12 $ factors. They are (1,3,9,27) and (5*1,5*3,5*9, 5*27) and (15*1,25*3,25*9, 25*27).
So if we are told that a number has exactly 8 factors: 8 = 1*8 or 2*4 or 2*2*2 it is either $p^7$ or $p*q^3$ or $p*q*s$. If we are told further that only of these factors is $75=3^3*5^2$ we can see this is impossible.
If a positive integer $a$ is factorized as the product of powers of distinct primes $$ a = p_{1}^{e_{1}} \cdots p_{n}^{e_{n}}, $$ then the number of positive factors of $a$ is $$\tag{numdiv} (e_{1} + 1) \cdot \dots \cdot (e_{n} + 1). $$ You know that $3^{3}$ and $5^{2}$ divide $a$. So you have at least $4 \cdot 3 = 12$ positive factors.
Concerning the lcm, if $27$ and $75$ divide $a$, then their least common multiple $675$ divides $a$. But by the formula (numdiv), $675$ has $12$ positive factors (which are $1, 3, 9, 27, 5, 15, 45, 135, 25, 75, 225, 675$), and these are all factors of $a$.