What is the largest 3digit number that has only 2 and 5 as p

What is the largest 3-digit number that has only 2 and 5 as prime factors?

Solution

We need to find the largest 3 digit number with only 2 and 5 as prime factors.

The number can be expressed as 2^n*5^m. 2^n*5^m should be less than 1000.

2^2*5^2 = 4*25 = 100

It is possible to multiply 100 by 5 only once and the result in that case is 500. So instead multiply 100 by a power of 2. The maximum power of 2 that this can be done with is 8.

100*8 = 800

800 = 2^5*5^2 and only has the prime factors 2 and 5. Multiplying this by either 2 or 5 gives a result that has 4 digits.

The largest 3 digit number with only 2 and 5 as prime factors is 800.