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Prime factors OG Quant #144

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Prime factors OG Quant #144

by wdgolden » Tue Aug 17, 2010 11:01 am
For any integer greater than 1, Ln denotes the product of all integers from 1 to n, inclusive. How many prime numbers are there between L6 + 2 and L6 + 6, inclusive? (Note, the "L" is used as a symbol in the original question.)

(A) None
(B) One
(C) Two
(D) Three
(E) Four

I get the long way of getting 6! + 2 and 6! + 6, and then determining primes between them. The book has a shorter way that I'm hoping someone can figure out and explain.
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by uwhusky » Tue Aug 17, 2010 11:12 am
Is the answer A?

Multiple of x + multiple of x will be divisible by x.

3(2) + 3 = divisible by 3.

So 6! = multiple of 2, 3, 4, 5, 6; and +2 --> +6 are all within those multiples. So any of those sum should result with the number being divisible, and thus there isn't any prime between the them?

Hopefully I am not wrong on this, or I might have to go on a drinking binge.
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by kvcpk » Tue Aug 17, 2010 11:18 am
uwhusky wrote:Is the answer A?

Multiple of x + multiple of x will be divisible by x.

3(2) + 3 = divisible by 3.

So 6! = multiple of 2, 3, 4, 5, 6; and +2 --> +6 are all within those multiples. So any of those sum should result with the number being divisible, and thus there isn't any prime between the them?

Hopefully I am not wrong on this, or I might have to go on a drinking binge.
You dont have to. :)
I think you are right!!
"Once you start working on something,
don't be afraid of failure and don't abandon it.
People who work sincerely are the happiest."
Chanakya quotes (Indian politician, strategist and writer, 350 BC-275BC)
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by uwhusky » Tue Aug 17, 2010 11:21 am
If I am wrong, I guess we're both wrong...and I have to buy drinks =).
Last edited by uwhusky on Tue Aug 17, 2010 11:22 am, edited 1 time in total.
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by selango » Tue Aug 17, 2010 11:22 am
L6=(1.2.3.4.5.6)

L6+2=2+(1.2.3.4.5.6)=2[1+(1.3.4.5.6)]-->Multiple of 2.Not prime

L6+3=3+(1.2.3.4.5.6)=3[1+(1.2.4.5.6)]-->Multiple of 3.Not prime

L6+4=4+(1.2.3.4.5.6)=4[1+(1.2.3.5.6)]-->Multiple of 4.Not prime

L6+5=5+(1.2.3.4.5.6)=5[1+(1.2.3.4.6)]-->Multiple of 5.Not prime

L6+6=6+(1.2.3.4.5.6)=6[1+(1.2.3.4.5)]-->Multiple of 6.Not prime

No prime between L6+2 and L6+6

Note that we don't repeat all the steps.Just from step 1 itself we can deduce the rem steps.

Pick A
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by wdgolden » Tue Aug 17, 2010 11:37 am
uwhusky - you're right, answer is (A). I wasn't familiar with
Multiple of x + multiple of x will be divisible by x.
. Is that a named rule I should know?

selengo - I take it you're factoring out a "2" in the first step you show?

L6+2 = 2 + (1 x 2 x 3 x 4 x 5 x 6) = 2 + 2(1 x 3 x 4 x 5 x 6) = 2[1 + 1(1 x 3 x 4 x 5 x 6)] = 2(1 + 3 x 4 x 5 x 6)
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by uwhusky » Tue Aug 17, 2010 11:43 am
wdgolen -

Yes, that's a rule you will need to remember. Selango showed in his equations on the proof of such rule.
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