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

The correct option is (b) - 111.

[as per provisional answerkey]

Solution

We are given the equation: ${10}^{m}x1000xn={75}^{25}x{25}^{32}x{32}^{75}$.
Since $1000={10}^3$, the left side is: ${10}^{(m+3)}xn$.
We are told that n is not divisible by 10. This means that in the prime factorization of the right-hand side, all possible pairs of $(2x5)$ must be contained within the 10^(m+3) term. Therefore, (m+3) must be equal to the total number of trailing zeros, or the exponent of 10, in the product on the right.

Step 1: Prime Factorization of the right-hand side
${75}^{25}=(3x{5}^2{)}^{25}=3^{25}x{5}^{50}$
${25}^{32}=(5^2{)}^{32}=5^{64}$
${32}^{75}=(2^5{)}^{75}=2^{375}$

Step 2: Combine the factors
Right-hand side = $3^{25}x{5}^{50}x{5}^{64}x{2}^{375}$
Right-hand side = $3^{25}x{5}^{(50+64)}x{2}^{375}$
Right-hand side = $3^{25}x{5}^{114}x{2}^{375}$

Step 3: Determine the power of 10
A power of 10 is formed by the pair $(2x5)$. The number of such pairs is determined by the lower exponent of 2 or 5.
Number of 5s = 114
Number of 2s = 375
Max pairs of $(2x5)$ = 114.
So, the expression can be written as: $(2x5{)}^{114}x{2}^{(375-114)}x{3}^{25}$
= ${10}^{114}x(2^{261}x{3}^{25})$

Step 4: Solve for m
Comparing ${10}^{(m+3)}xn={10}^{114}x(2^{261}x{3}^{25})$:
Since $n=2^{261}x{3}^{25}$ (which is not divisible by 10 because it contains no factors of 5), we have:
$m+3=114$
$m=114-3$
$m=111$

Why the other options are incorrect

• Option (a) - 101: This result would occur if the total exponent of 10 was calculated as 104 instead of 114, likely by miscalculating the power of 5 in 75^25 or 25^32.
• Option (c) - 121: This value might be reached if one fails to subtract the 3 from the 1000 (10^3) term or adds it instead of subtracting it from the total power of 114.
• Option (d) - 131: This value is significantly higher and would require a much larger count of factor 5 in the prime factorization, which is not supported by the given numbers.

Key Concept

The exponent of 10 in a product is determined by the minimum of the exponents of its prime factors 2 and 5 (the "limiting factor" principle in prime factorization).