Question

If m and n are natural numbers such that n > 1, and mⁿ = 2²⁵ × 3⁴⁰, then m − n equals

• A. 209942
• B. 209947
• C. 209932
• D. 209937

Answer 1

The Correct Option is B

To find the value of \( m - n \) where \( m \) and \( n \) are natural numbers, \( n > 1 \), and \( m^n = 2^{25} \times 3^{40} \), follow these steps:

1. Express \( m \) in terms of prime factors:
   \( m = 2^a \times 3^b \) 
2. Substitute \( m \) into the equation:
   \( (2^a \times 3^b)^n = 2^{25} \times 3^{40} \)
3. Apply the power rule for exponents \( (x^m)^n = x^{mn} \):
   \( 2^{an} \times 3^{bn} = 2^{25} \times 3^{40} \)
4. Equate the powers of similar bases:
   \( an = 25 \),
   \( bn = 40 \)
5. Solve for \( a \) and \( b \):
   \( a = \frac{25}{n} \),
   \( b = \frac{40}{n} \)
6. Since \( a \) and \( b \) must be integers, \( n \) must divide both 25 and 40.
   GCD of 25 and 40 is 5, so \( n = 5 \)
7. Substitute \( n = 5 \) into equations:
   \( a = \frac{25}{5} = 5 \),
   \( b = \frac{40}{5} = 8 \)
8. Calculate \( m \):
   \( m = 2^5 \times 3^8 = 32 \times 6561 = 209952 \)
9. Determine \( m - n \):
   \( 209952 - 5 = \mathbf{209947} \)

Thus, the value of \( m - n \) is 209947.