If $10^m \times 1000 \times n = 75^{25} \times 25^{32} \times 32^{75}$, where $n$ is not divisible by $10$, then the value of $m$ is
Detailed Explanation:
Prime factorize the RHS: $75^{25} = (3 \times 5^2)^{25} = 3^{25} \times 5^{50}$, $25^{32} = 5^{64}$, and $32^{75} = 2^{375}$. Combined: $2^{375} \times 3^{25} \times 5^{114}$.
Maximum power of 10: Since $10 = 2 \times 5$, the limiting factor is $\min(375, 114) = 114$. Thus, RHS = $10^{114} \times 2^{261} \times 3^{25}$, where $(2^{261} \times 3^{25})$ is not divisible by $10$ (no factor of $5$).
LHS comparison: $10^m \times 1000 \times n = 10^{m+3} \times n$. Since $n$ is not divisible by $10$, we equate: $m + 3 = 114 \Rightarrow m = 111$.
Therefore, m = 111, making Option 2 the correct answer.
Question 2 of 7 HCF and LCM
Practice PYQ questions from this topic across all years
$X$, $Y$ and $Z$ jump forward $4'$, $6'$ and $5'$, respectively. At 8 AM, they all la...
If the product of the HCF and LCM of two distinct numbers is the cube of one of the n...