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An integer n that is greater than 1 is said to be "prime-sat

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An integer n that is greater than 1 is said to be "prime-sat  [#permalink]

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New post 17 Sep 2013, 00:16
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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 \(\sqrt{n}\). Which of the following integers is prime saturated?

A) 6
B) 35
C) 46
D) 66
E) 75

Can someone explain the answer choices. Thanks!

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Re: An integer n that is greater than 1 is said to be "prime-sat  [#permalink]

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New post 17 Sep 2013, 00:34
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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 \(\sqrt{n}\). Which of the following integers is prime saturated?

A. 6
B. 35
C. 46
D. 66
E. 75

According to the definition integer is said to be "prime-saturated" if the square of its largest prime is smaller than the integer itself.

Only E satisfies this: 75=3*5^2 --> the largest prime is 5 --> 5^2=25<75.

Answer: E.

Similar question to practice: a-number-is-said-to-be-prime-saturated-106511.html

Hope it helps.
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An integer n that is greater than 1 is said to be "prime-sat  [#permalink]

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New post 29 Jun 2015, 10:41
Bunuel 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 \(\sqrt{n}\). Which of the following integers is prime saturated?

A. 6
B. 35
C. 46
D. 66
E. 75

According to the definition integer is said to be "prime-saturated" if the square of its largest prime is smaller than the integer itself.

Only E satisfies this: 75=3*5^2 --> the largest prime is 5 --> 5^2=25<75.

Answer: E.

Similar question to practice: a-number-is-said-to-be-prime-saturated-106511.html

Hope it helps.



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Re: An integer n that is greater than 1 is said to be "prime-sat  [#permalink]

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New post 11 Jan 2017, 03:23
Great Question.
Here is what i did in this one =>
If n is to be a Saturated prime -> All prime factors of n must be less than equal to √n
Start with option E=>
75=> √75=8.something
75=>3*5^2
Clearly both 3 and 5 are less than 8.something
Hence E.

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Re: An integer n that is greater than 1 is said to be "prime-sat  [#permalink]

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New post 16 Jan 2017, 16:51
1
fozzzy 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 \(\sqrt{n}\). Which of the following integers is prime saturated?

A) 6
B) 35
C) 46
D) 66
E) 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
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Re: An integer n that is greater than 1 is said to be "prime-sat  [#permalink]

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New post 30 Jan 2018, 07:35
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fozzzy 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 \(\sqrt{n}\). Which of the following integers is prime saturated?

A) 6
B) 35
C) 46
D) 66
E) 75

Can someone explain the answer choices. Thanks!


So, n is prime saturated if the largest prime factor of n < √n.
If we square both sides of the inequality, we get: n is prime saturated if (the largest prime factor of n)² < n

A. 6 = (2)(3) ---> 3² > 6 n is NOT PRIME SATURATED
B. 35 = (5)(7) ---> 7² > 35 n is NOT PRIME SATURATED
C. 46 = (2)(23) ---> 23² > 46 n is NOT PRIME SATURATED
D. 66 = (2)(3)(11) ---> 11² > 66 n is NOT PRIME SATURATED
E. 75 = (3)(5)(5) ---> 5² < 75 n IS PRIME SATURATED

Answer: E

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Re: An integer n that is greater than 1 is said to be "prime-sat &nbs [#permalink] 30 Jan 2018, 07:35
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