Question

If the number of positive divisors of the number $N=2^4 \times 3^7 \times 5^9 \times 7^9$ which are perfect square is $\lambda$. Then $\frac{\lambda}{10}$ equals to

Answer 1

30

Answer 2

A divisor $d$ of $N=2^4 \times 3^7 \times 5^9 \times 7^9$ is of the form $2^a \times 3^b \times 5^c \times 7^e$, where $0 \le a \le 4$, $0 \le b \le 7$, $0 \le c \le 9$, and $0 \le e \le 9$. For $d$ to be a perfect square, the exponents $a, b, c, e$ must be even integers. The number of possible even values for each exponent are: For $a$: $\{0, 2, 4\}$ (3 options) For $b$: $\{0, 2, 4, 6\}$ (4 options) For $c$: $\{0, 2, 4, 6, 8\}$ (5 options) For $e$: $\{0, 2, 4, 6, 8\}$ (5 options) The total number of divisors that are perfect squares is $\lambda = 3 \times 4 \times 5 \times 5 = 300$. The required value is $\frac{\lambda}{10} = \frac{300}{10} = 30$.