Question

Let \(a, b, m\) and \(n\) be natural numbers such that \(a >1\) and \(b >1\) . If \(a^m+b^n = 144^{145}\) , then the largest possible value of \(n − m\) is

• A. 579
• B. 289
• C. 580
• D. 290

Answer 1

The Correct Option is A

Given: 
We are provided with the equation: \(a^m \times b^n = 144^{145}\) where \(a > 1\) and \(b > 1\).
Also, we know that 144 can be expressed as: \(144 = 2^4 \times 3^2\).
Therefore, the equation can be rewritten as:

\(a^m \times b^n = 144^{145} = (2^4 \times 3^2)^{145}\)

By expanding the powers, we get: \(a^m \times b^n = 2^{580} \times 3^{290}\)

Step-by-Step Analysis:

1. Identifying the form of the equation:
   The equation \(a^m \times b^n = 2^{580} \times 3^{290}\) suggests that the powers of the prime factors of \(a\) and \(b\) must match the powers of 2 and 3, respectively. So we can make the following observations:
   • One of the terms must involve the prime factor 2, and the other must involve the prime factor 3.
   • Thus, we conclude that \(a^m\) must be a power of 2, and \(b^n\) must be a power of 3.
2. Considering the term involving \(a^m\):
   From the equation, since \(a^m\) must be a power of 2, we can write: \(a^m = 2^{580}\). This means that \(a\) is a power of 2. Therefore, \(a = 2\) and \(m = 580\).
3. Considering the term involving \(b^n\):
   Similarly, \(b^n\) must be a power of 3, so: \(b^n = 3^{290}\). This means that \(b\) is a power of 3. Therefore, \(b = 3\) and \(n = 290\).
4. Determining the smallest value of \(m\) and largest value of \(n\):
   As per the question, we need to find the least possible value of \(m\) and the largest possible value of \(n\). We know that:
   • The least possible value of \(m\) is 1, since \(a = 2\) and the minimum exponent for \(2^1\) would make \(m = 1\).
   • The largest possible value of \(n\) is 580, since \(b = 3\) and \(b^n = 3^{580}\).
5. Final calculation of \(n - m\):
   The difference \(n - m\) is: \(n - m = 580 - 1 = 579\)

Conclusion:
Therefore, the correct answer is (A): 579.