I have a number given in terms of its prime factors: $2^{97} * 3^{48} * 5^{24} * 7^{16}$. I have to form the largest "base" (number) from these numbers with the condition that the selection must itself be at least a power of 10, i.e. the power of the base must be greater than or equal to 10.
Valid combinations would be $2^{97}$, $(2*3)^{48}$, $(2*5)^{24}$ and so on. As long as the exponent is at least 10. What is the largest such number I can make? My intuition tells me it should be $(2^{6}*3^{3}*5*7)$ because those are the highest powers of the given primes that can themselves be written as powers of 16. But apparently, the correct answer is $(2^{9} * 3^{4} * 5^{2} * 7)$... I'm not disputing the validity of the answer, because obviously it's correct. But how do I figure this out analytically or algorithmically? Must I do a range search?
You have to find out the largest multiples of $10$ not exceeding the given exponents, in this case $90,40,20,10$. This gives $$2^{90}\cdot 3^{40}\cdot 5^{20}\cdot 7^{10}=(2^9\cdot 3^4\cdot 5^2\cdot 7)^{10}$$
Larger exponents for the base are not possible because they would be multplied with $10$ or more and lead to an exponent being too large.