If for any two natural numbers a and b, ab = 125, then ba is

When a number N can be written as ab for natural numbers a and b, N is expressed as a prime raised to a power: if N = pk for a prime p, then a = p and b = k is a natural-number solution — matching exponents is the general method.

1. Factorize 125 into primes: 125 = 5 × 5 × 5 = 53.

2. Match this to the form ab: comparing 53 with ab gives a = 5 and b = 3.

3. Substitute into ba: ba = 35.

4. Compute the product: 35 = 3 × 3 × 3 × 3 × 3 = 243.

• Cross-check: 53 = 125, matching the given equation, so a = 5, b = 3 is confirmed.

• The other technically valid pair, a = 125 and b = 1, gives ba = 1125 = 1, which is not among the offered options, so it is not the intended solution.

So ba = 243.