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)-Â1 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.
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