Question

Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive intefral divisor including 1 itself. Find n/95

Answer 1

Answer:

don't know bro...........,..............................................

Answer 2

Answer:

The smallest number is 432

Step-by-step explanation:

• The prime factorization of $75=3^{1} 5^{2}=(2+1)(4+1)(4+1)$.
• For $n$ to have exactly 75 integral divisors, we need to have $n=p_{1}^{e_{1}-1} p_{2}^{e_{2}-1} \cdots$ such that $e_{1} e_{2} \cdots=75$. Since $75 \mid n$, two of the prime factors must be 3 and 5 .
• To minimize $n$, we can introduce a third prime factor, 2 . Also to minimize $n$, we want 5 , the greatest of all the factors, to be raised to the least power. Therefore, $n=2^{4} 3^{4} 5^{2}$ and $\frac{n}{75}=\frac{2^{4} 3^{4} 5^{2}}{3 \cdot 5^{2}}\\=16 \cdot 27\\=432$