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The smallest prime factor of both a and b is 7. What is the smallest p

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nick1816
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The smallest prime factor of both a and b is 7. What is the smallest p [#permalink] New post   12 Nov 2019, 20:30
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The smallest prime factor of both a and b is 7. What is the smallest prime factor of a+b?

A. 7
B. 5
C. 3
D. 2
E. Can't be determined
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D
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globaldesi
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Re: The smallest prime factor of both a and b is 7. What is the smallest p [#permalink] New post   12 Nov 2019, 20:43
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Let a = 7*11
b = 7*13
now a+b = 7(11+13)
=7(24)
= 7*2*2*2*3
smallest =2
also consider a = b
in that case also smallest will be 2
Hence D
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rajatchopra1994
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The smallest prime factor of both a and b is 7. What is the smallest p [#permalink] New post   Updated on: 13 Nov 2019, 03:20
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nick1816 wrote:
The smallest prime factor of both a and b is 7. What is the smallest prime factor of a+b?

A. 7
B. 5
C. 3
D. 2
E. Can't be determined


As 7 is a prime factor of a & b
therefore, a & b should be greater than or equal to 7.
.i.e. a≥7, b≥7

Now consider a case in a=7, b=7, so a+b= 14.
So the smallest prime factor is 2 (of a+b).

let us consider another case, a= 7^2 ,b=7^2.
so a+b= 98 which is even number. And again 2 is the smallest prime factor.

Similarly for all values a^n + b^n divisible by.
Thus, correct option is D.

Pls give Kudos :thumbup: :thumbup:

Originally posted by rajatchopra1994 on 12 Nov 2019, 22:17.
Last edited by rajatchopra1994 on 13 Nov 2019, 03:20, edited 1 time in total.
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Re: The smallest prime factor of both a and b is 7. What is the smallest p [#permalink] New post   13 Nov 2019, 00:22
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We can consider 3 scenarios since a and b are multiples of 7.

Scenario 1: a= 7; b=14. Smallest common prime factor is 7.

A+b= 21, which is 7x3. Hence smallest prime factor is 3

Scenario 2: a= 14; b= 21. Common smallest prime factor is again 7

a+b= 35. Smallest prime factor is 5

Scenario 3:

As told above: a= 7x11 b= 7x 13

Smallest prime factor of a+b=2.

Hence answer is E: can’t be determined.
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Re: The smallest prime factor of both a and b is 7. What is the smallest p [#permalink] New post   17 Nov 2019, 20:19
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nick1816 wrote:
The smallest prime factor of both a and b is 7. What is the smallest prime factor of a+b?

A. 7
B. 5
C. 3
D. 2
E. Can't be determined


Since the smallest prime factor of a and b is 7, both a and b must be odd multiples of 7 (notice that if a or b is an even multiple of 7, say 14, then 2, not 7, would be the smallest prime factor). Thus, the sum of a and b is even and so the smallest prime factor of the sum of a and b is 2.

Answer: D
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Re: The smallest prime factor of both a and b is 7. What is the smallest p [#permalink] New post   07 Feb 2024, 09:27
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Re: The smallest prime factor of both a and b is 7. What is the smallest p [#permalink]
07 Feb 2024, 09:27
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