If m and n are natural numbers such that n > 1, and m^n = 2^15 × 3^10, then the value of m − n is.
Answer and solution
Correct answer: 67
Solution
Answer: 67
1. We are given:
2. mⁿ = 2¹⁵ × 3¹⁰
3. Since m and n are natural numbers, the exponent n must divide both 15 and 10.
4. The only common factor of 15 and 10 that is greater than 1 is 5.
5. Therefore, n = 5.
6. Now take the fifth root:
7. m = 2³ × 3²
m = 8 × 9
m = 72
8. Hence,
9. m − n = 72 − 5 = 67.
10. Answer: 67 (TITA).
Why this method works:
In a perfect nth power, every prime exponent is divisible by n. Here n must divide gcd(15,10)=5. The condition n>1 eliminates 1, leaving n=5 uniquely; no other prime factors can appear in m.
Independent check and common trap:
With m=72=2³×3², raising to the fifth power gives precisely 2¹⁵×3¹⁰. The final quantity is m−n, so subtract 5 from 72 after finding the base, rather than stopping at either intermediate value.
Apply: n divides both 15 and 10, with n>1: n=5. m=2³×3²=72; m−n=67.
Check: With m=72=2³×3², raising to the fifth power gives precisely 2¹⁵×3¹⁰. The final quantity is m−n, so subtract 5 from 72 after finding the base, rather than stopping at either intermediate value.
1. We are given:
2. mⁿ = 2¹⁵ × 3¹⁰
3. Since m and n are natural numbers, the exponent n must divide both 15 and 10.
4. The only common factor of 15 and 10 that is greater than 1 is 5.
5. Therefore, n = 5.
6. Now take the fifth root:
7. m = 2³ × 3²
m = 8 × 9
m = 72
8. Hence,
9. m − n = 72 − 5 = 67.
10. Answer: 67 (TITA).
Why this method works:
In a perfect nth power, every prime exponent is divisible by n. Here n must divide gcd(15,10)=5. The condition n>1 eliminates 1, leaving n=5 uniquely; no other prime factors can appear in m.
Independent check and common trap:
With m=72=2³×3², raising to the fifth power gives precisely 2¹⁵×3¹⁰. The final quantity is m−n, so subtract 5 from 72 after finding the base, rather than stopping at either intermediate value.
G Strategy shortcut
Spot: Number System – Exponents.Apply: n divides both 15 and 10, with n>1: n=5. m=2³×3²=72; m−n=67.
Check: With m=72=2³×3², raising to the fifth power gives precisely 2¹⁵×3¹⁰. The final quantity is m−n, so subtract 5 from 72 after finding the base, rather than stopping at either intermediate value.
G Strategy: Spot: Number System – Exponents. Apply: n divides both 15 and 10, with n>1: n=5. m=2³×3²=72; m−n=67. Check: With m=72=2³×3², raising to the fifth power gives precisely 2¹⁵×3¹⁰. The final quantity is m−n, so subtract 5 from 72 after finding the base, rather than stopping at either intermediate value.