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Math Expert V
Joined: 02 Sep 2009
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If a and b are prime numbers, such that a > b, which of the following  [#permalink]

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If a and b are prime numbers, such that a > b, which of the following cannot be true?

A. a+b is prime.
B. ab is odd.
C. a(a-b) is odd.
D. a-b is prime.
E. a^b is even.

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Posts: 3554
If a and b are prime numbers, such that a > b, which of the following  [#permalink]

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Bunuel wrote:
If a and b are prime numbers, such that a > b, which of the following cannot be true?

A. a+b is prime.
B. ab is odd.
C. a(a-b) is odd.
D. a-b is prime.
E. a^b is even.

Answer E. $$a$$ must be odd. All primes except 2 are odd. And $$a$$ must be greater than $$b$$. Thus the only even prime, 2, cannot be $$a$$.

An odd base raised to any integral power (even or odd) -- is odd. The expression $$a^{b}$$ cannot be even.

Because $$a > b$$, the only even power, 2, cannot be "a." There are no prime numbers smaller than 2.

So $$a$$ is odd. Its minimum possible value is 3, and all primes greater than 3 are odd.

Rule: An odd base raised to any integer power is always odd. (It is probably easier simply to remember that odd times odd equals odd, and extend that principle.)

Whether an odd number is raised to a power of 2, 3, 4, 5, or 51, the result is odd * odd (* odd * odd . . .) for as many powers as there are. The exponent's even or odd sign is irrelevant. Examples:

If $$a = 3, b = 2$$, then $$3^2 = 9$$, and odd * odd = odd

If $$a = 5, b = 3$$, then $$5^3 = 5 * 5 * 5 = 125$$

Other answer choices can be disproved. Which of following cannot be true?

A. $$a+b$$ is prime.
If $$a = 3, b = 2$$, $$(3 + 2) = 5$$. 5 is prime. It can be true. REJECT

B. $$ab$$ is odd.
If $$a = 5, b = 3$$, $$(ab) = 15$$, which is odd. Can be true. REJECT

C. $$a(a-b)$$ is odd.
If $$a = 3, b = 2$$: $$(3)(3-2) = (3)(1) = 3$$, which is odd. Can be true. REJECT

D. $$a-b$$ is prime.
If $$a = 5, b = 3$$, $$(5 - 3) = 2$$, which is prime. Can be true. REJECT

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Re: If a and b are prime numbers, such that a > b, which of the following  [#permalink]

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3^2 = 9; 5^3 = 125; 7^2 = 49

The list goes on, but just a little evidence to show E is the correct answer.

By POE:

If a and b are prime numbers, such that a > b, which of the following cannot be true?

A. a+b is prime. 5+2 = 7 Prime
B. ab is odd. 5*7 = 35 Odd
C. a(a-b) is odd. 3(3-2) = 3 Odd
D. a-b is prime. 7-5 = 2 Prime
E. a^b is even. Answer
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Re: If a and b are prime numbers, such that a > b, which of the following  [#permalink]

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genxer123 wrote:
Bunuel wrote:
If a and b are prime numbers, such that a > b, which of the following cannot be true?

A. a+b is prime.
B. ab is odd.
C. a(a-b) is odd.
D. a-b is prime.
E. a^b is even.

Answer E. $$a$$ must be odd. All primes except 2 are odd. And $$a$$ must be greater than $$b$$. Thus the only even prime, 2, cannot be $$a$$.

An odd base raised to any integral power (even or odd) -- is odd. The expression $$a^{b}$$ cannot be even.

Because $$a > b$$, the only even power, 2, cannot be "a." There are no prime numbers smaller than 2.

So $$a$$ is odd. Its minimum possible value is 3, and all primes greater than 3 are odd.

Rule: An odd base raised to any integer power is always odd. (It is probably easier simply to remember that odd times odd equals odd, and extend that principle.)

Whether an odd number is raised to a power of 2, 3, 4, 5, or 51, the result is odd * odd (* odd * odd . . .) for as many powers as there are. The exponent's even or odd sign is irrelevant. Examples:

If $$a = 3, b = 2$$, then $$3^2 = 9$$, and odd * odd = odd

If $$a = 5, b = 3$$, then $$5^3 = 5 * 5 * 5 = 125$$

Other answer choices can be disproved. Which of following cannot be true?

A. $$a+b$$ is prime.
If $$a = 3, b = 2$$, $$(3 + 2) = 5$$. 5 is prime. It can be true. REJECT

B. $$ab$$ is odd.
If $$a = 5, b = 3$$, $$(ab) = 15$$, which is odd. Can be true. REJECT

C. $$a(a-b)$$ is odd.
If $$a = 3, b = 2$$: $$(3)(3-2) = (3)(1) = 3$$, which is odd. Can be true. REJECT

D. $$a-b$$ is prime.
If $$a = 5, b = 3$$, $$(5 - 3) = 2$$, which is prime. Can be true. REJECT

But 2^5=32. Then E can be true.
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Joined: 12 Sep 2015
Posts: 4007
If a and b are prime numbers, such that a > b, which of the following  [#permalink]

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Top Contributor
Rupanjel wrote:
But 2^5=32. Then E can be true.

Be careful.
In your counter-example (2^5), you have a = 2 and b = 5. This breaks the given condition that a > b

Cheers,
Brent
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Re: If a and b are prime numbers, such that a > b, which of the following  [#permalink]

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GMATPrepNow wrote:
Rupanjel wrote:
But 2^5=32. Then E can be true.

Be careful.
In your counter-example (2^5), you have a = 2 and b = 5. This breaks the given condition that a > b

Cheers,
Brent

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Re: If a and b are prime numbers, such that a > b, which of the following  [#permalink]

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Bunuel wrote:
If a and b are prime numbers, such that a > b, which of the following cannot be true?

A. a+b is prime.
B. ab is odd.
C. a(a-b) is odd.
D. a-b is prime.
E. a^b is even.

Recall that all the prime numbers other than 2 are odd. Since a > b, a must be odd (even if b = 2). Thus, a^b must be odd since any odd number raised to a power is odd. So, a^b can’t be even.

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Re: If a and b are prime numbers, such that a > b, which of the following  [#permalink]

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_________________ Re: If a and b are prime numbers, such that a > b, which of the following   [#permalink] 17 Oct 2018, 00:36
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