Problem Solving

For an odd integer n, the function f(n) is defined as the product of all odd integers from 1 to n. The lowest odd prime factor f(71)-­1 lies between...

A. 3 and 10
B. 11 and 30
C. 31 and 50
D. 51 and 70
E. 71 and above

I'm confused how to set up the formulas here. Can any experts help?

ardz24 wrote:For an odd integer n, the function f(n) is defined as the product of all odd integers from 1 to n. The lowest odd prime factor f(71)-­1 lies between...

A. 3 and 10
B. 11 and 30
C. 31 and 50
D. 51 and 70
E. 71 and above

Since the difference between them is 1, f(71) and f(71)-1 are consecutive integers.
Consecutive integers are COPRIMES: they share no factors other than 1.

Let's examine why:

If x is a multiple of 2, then the next smallest multiple of 2 is x-2.
If x is a multiple of 3, then the next smallest multiple of 3 is x-3.
If x is a multiple of 4, then the next smallest multiple of 4 is x-4.

Using this logic, if we go from x to x-1, we get only to the next smallest multiple of 1.
Implication:
1 is the only factor that x and x-1 have in common.
In other words, x and x-1 are COPRIMES.

Thus:
f(71) and f(71)-1 are COPRIMES.
They share no factors other than 1.

f(71) = 1 * 3 * 5 *....* 67 * 69 * 71.

Looking at the values on the right, we can see that every odd prime number between 1 and 71, inclusive, is a factor of f(71).
Since f(71) and f(71)-1 are coprimes, NONE of the odd prime numbers between 1 and 71, inclusive, can be a factor of f(71)-1.

Thus, the smallest odd prime factor of f(71)-1 must be GREATER THAN 71.

The correct answer is E.