Question:medium

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

Updated On: Mar 20, 2026
  • 209942
  • 209947
  • 209932
  • 209937
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The Correct Option is B

Solution and Explanation

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

  1. Represent \( m \) using its prime factors:
    \( m = 2^a \times 3^b \)
  2. Substitute this expression for \( m \) into the given equation:
    \( (2^a \times 3^b)^n = 2^{25} \times 3^{40} \)
  3. Apply the exponent rule \( (x^p)^q = x^{pq} \):
    \( 2^{an} \times 3^{bn} = 2^{25} \times 3^{40} \)
  4. Equate the exponents for each prime base:
    \( an = 25 \),
    \( bn = 40 \)
  5. Solve for \( a \) and \( b \) in terms of \( n \):
    \( a = \frac{25}{n} \),
    \( b = \frac{40}{n} \)
  6. Since \( a \) and \( b \) must be integers, \( n \) must be a common divisor of 25 and 40. The greatest common divisor (GCD) of 25 and 40 is 5. Since \( n>1 \), \( n \) must be 5.
    Therefore, \( n = 5 \)
  7. Substitute \( n = 5 \) back into the equations for \( a \) and \( b \):
    \( a = \frac{25}{5} = 5 \),
    \( b = \frac{40}{5} = 8 \)
  8. Calculate the value of \( m \):
    \( m = 2^5 \times 3^8 = 32 \times 6561 = 209952 \)
  9. Calculate \( m - n \):
    \( 209952 - 5 = \mathbf{209947} \)

The value of \( m - n \) is 209947.

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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
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