tejal777 wrote:
Integer x is equal to the product of all even numbers from 2 to 60. If y is the smallest prime no. that is also a factor of x-1, then which of the following expressions must be true?
y lies bet. 0 and 4
y lies bet. 4 and 10
y lies bet. 10 and 20
y lies bet. 20 and 30
y is greater than 30
Guys please help me understand the underlying principles..
This is a replication of one of the most discussed and hardest question (I guess the question is from retired gmat question bank).
Per the question, x = the product of all even numbers from 2 to n, where n is any even integer value from 2 to 60 including.
If x = 2, x - 1 = 2! -1
If x = 2x4, x - 1 = 2! (2^2) - 1
If x = 2x4x6, x - 1 = 3! (2^3) - 1
If x = 2x4x6x8, x - 1 = 4! (2^4) - 1
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If x = 2x4x6x.......x60, x - 1 = 30! (2^30) - 1
Lets find y, a smallest prime factor of (x-1):
If y were 2, [(30!) (2^30) -(1)] would be divisible by 2 however thats not the case.
If y were 3, [(30!) (2^30) -(1)] would be divisible by 3 however thats not the case.
If y were 5, [(30!) (2^30) -(1)] would be divisible by 5 however thats not the case.
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If y were 29, [(30!) (2^30) -(1)] would be divisible by 29 however thats not the case.
If none of the primes under 31 is a factor of (x-1), (x-1) must have a prime factore above 30.
Thats E.
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