I was going over some old maths competition questions for primary school children. The hardest question on the paper was this...
The number $2000 = 2^4 \times 5^3$ is the product of seven prime factors. Let x be the smallest integer greater than 2000 with this property and y be the largest integer smaller than 2000 with this property. What is the value of x - y?
I know this has to do with the $\Omega(n)$ function. I guess I know
- $\Omega(x) = 7$
- $\Omega(y) = 7$
- $x > 2000$ and
- $y < 2000$
Other than that I have absolutely no idea how to solve this other than to write an algorithm.
Any suggestions would be greatly appreciated. Thanks in advance.
I don't know of an elegant way to find the answer, but let me present the answer for reference.
I used brute force. Factorizing the integers $1999$, $1998$, and so on, we find that the largest integer less than $2000$ that has exactly seven prime factors is $1984=2^6\times 31$. Factorizing the integers $2001$, $2002$, and so on, we find that the smallest integer greater than 2000 that has exactly seven prime factors is $2080=2^5\times 5\times 13$. (Note that $2016=2^5\times 3^2\times 7$ has exactly eight prime factors, and that $2048=2^{11}$ has exactly eleven prime factors.)
Thus, $x=2080$, $y=1984$, and $x-y=96$.