Problem Solving

sana.noor wrote:An integer n that is greater than 1 is said to be "prime saturated" if it has no prime factor greater than or equal to square root of n. which of the following integers is prime-saturated?

In other words, n will be prime saturated if the square of largest prime factor of n is smaller than n. Let's check the options one by one...

• A. 6 = 2*3 ---> 3² > 6 ---> NO
  B. 35 = 5*7 ---> 7² > 35 ---> NO
  C. 46 = 2*23 ---> 23² > 46 ---> NO
  D. 66 = 2*3*11 ---> 11² > 66 ---> NO
  E. 75 = 3*5*5 ---> 5² < 75 ---> YES

The correct answer is E.

sana.noor wrote:An integer n that is greater than 1 is said to be "prime saturated" if it has no prime factor greater than or equal to square root of n. which of the following integers is prime-saturated?
6
35
46
66
75

We will find the largest prime factor of each answer choice. If the largest prime factor of the number is greater than the square root of the number, then the number is NOT prime saturated, since by definition no prime factor of the number is greater than or equal to its square root.

Furthermore, instead of comparing the largest prime factor and the square root of the number, we will compare the square of the largest prime factor and square of the square root of the number, i.e., the number itself. That is because if two numbers, x and y, are positive, √x > √y implies x > y.

A) 6

The largest prime factor of 6 is 3. Since 3^2 = 9 is greater than (√6)^2 = 6, 6 is NOT prime saturated.

B) 35

The largest prime factor of 35 is 7. Since 7^2 = 49 is greater than (√35)^2 = 35, 35 is NOT prime saturated.

C) 46

The largest prime factor of 46 is 23. Since 23^2 is greater than (√46)^2 = 46, 46 is NOT prime saturated.

D) 66

The largest prime factor of 66 is 11. Since 11^2 = 121 is greater than (√66)^2 = 66, 66 is NOT prime saturated.

E) 75

The largest prime factor of 75 is 5. Since 5^2 = 25 is NOT greater than (√75)^2 = 75, 75 IS prime saturated.

Answer: E