GMAT Math

1) For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n inclusive. If p is the smallest prime factor of h(100) + 1, then p is between
a. 2 and 10
b. 10 and 20
c. 20 and 30
d. 30 and 40
e. > 40


2) Positive integers x, y, and z are such that x is a factor of y and y is a factor of z, if z even? (EXTRA DIFFICULT)
a. xz is even
b. y is even

According to the given function,
h(100) = 2*4*6*8*...*100

By factoring a 2 from each term of our function, h(100) can be rewritten as
2^50*(1*2*3*...*50).

Thus, all integers up to 50 - including all prime numbers up to 50 - are factors of h(100).

Therefore, h(100) + 1 cannot have any prime factors 50 or below, since dividing this value by any of these prime numbers will yield a remainder of 1.

Since the smallest prime number that can be a factor of h(100) + 1 has to be greater than 50, The correct answer is E. According to the given function,
h(100) = 2*4*6*8*...*100

By factoring a 2 from each term of our function, h(100) can be rewritten as
2^50*(1*2*3*...*50).

Thus, all integers up to 50 - including all prime numbers up to 50 - are factors of h(100).

Therefore, h(100) + 1 cannot have any prime factors 50 or below, since dividing this value by any of these prime numbers will yield a remainder of 1.

Since the smallest prime number that can be a factor of h(100) + 1 has to be greater than 50, The correct answer is E.

rehandossani wrote:1) For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n inclusive. If p is the smallest prime factor of h(100) + 1, then p is between
a. 2 and 10
b. 10 and 20
c. 20 and 30
d. 30 and 40
e. > 40


2) Positive integers x, y, and z are such that x is a factor of y and y is a factor of z, if z even? (EXTRA DIFFICULT)
a. xz is even
b. y is even

2nd is a lot easier than the 1st one.

x is factor of y => y = k1*x
y is a factor of z => z = k2*y

1) Z is even if y is even. so statement B is sufficient to answer z is even.

2) Statement A is xz = even;

z = k2y = k1k2x;

Given as xz is even; either x is even or z is even or both.

Assuming x is even-> As z = k1k2x; Therefore x is even.

If x is odd then z is even from Statement A.

So Answer is D (Both Stmts can independently answer the question)

Sid

rehandossani wrote:1) For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n inclusive. If p is the smallest prime factor of h(100) + 1, then p is between
a. 2 and 10
b. 10 and 20
c. 20 and 30
d. 30 and 40
e. > 40


2) Positive integers x, y, and z are such that x is a factor of y and y is a factor of z, if z even? (EXTRA DIFFICULT)
a. xz is even
b. y is even