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What is the greatest prime factor of 1+2+3+….+36?

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What is the greatest prime factor of 1+2+3+….+36?  [#permalink]

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New post 08 Oct 2018, 01:53
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[Math Revolution GMAT math practice question]

What is the greatest prime factor of \(1+2+3+….+36\)?

\(A. 2\)
\(B. 3\)
\(C. 9\)
\(D. 31\)
\(E. 37\)

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Re: What is the greatest prime factor of 1+2+3+….+36?  [#permalink]

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New post 08 Oct 2018, 02:15
Sum = n(n+1)/2. Sum = 36(37)/2 = 18*37.

Prime factoring gives 3^2*2*37.

Greatest prime factor = 37.

E is the answer

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Re: What is the greatest prime factor of 1+2+3+….+36?  [#permalink]

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New post 08 Oct 2018, 04:27
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the greatest prime factor of \(1+2+3+….+36\)?

\(A. 2\)
\(B. 3\)
\(C. 9\)
\(D. 31\)
\(E. 37\)


sum can be written as
36*37/2 = 37 * 18
greatest prime factor = 37
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Re: What is the greatest prime factor of 1+2+3+….+36?  [#permalink]

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New post 08 Oct 2018, 05:16
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the greatest prime factor of \(1+2+3+….+36\)?

\(A. 2\)
\(B. 3\)
\(C. 9\)
\(D. 31\)
\(E. 37\)



The challenge is that how we can determine the sum of this serious. we can do it by applying the following formula.

36 + 1 = 37

35 + 2 = 37

at the same way we can make such pair. we will have 18 pairs as total we have is 36 integers.

37*18 = total. NO need to multiply.

*** 37 is a prime. So it's the greatest prime.

The best answer is E.
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What is the greatest prime factor of 1+2+3+….+36?  [#permalink]

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New post 08 Oct 2018, 05:50
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the greatest prime factor of \(1+2+3+….+36\)?

\(A. 2\)
\(B. 3\)
\(C. 9\)
\(D. 31\)
\(E. 37\)

\(?\,\,\,:\,\,\,{\rm{greatest}}\,\,{\rm{prime}}\,\,{\rm{factor}}\)

\(1 + 2 + 3 + \ldots + 36\,\,\mathop = \limits^{\left( * \right)} \,\,\left( {{{1 + 36} \over 2}} \right) \cdot 36 = 18 \cdot 37\,\,\,\,\mathop \Rightarrow \limits^{2,3\,\,{\rm{or}}\,\,37} \,\,\,\,? = 37\)

(*) 1, 2, 3, ..., 36 is a finite arithmetic sequence, hence their average is equal to the average of the first and last terms (or any two symmetric to the median).
Besides that, the sum 1+2+3+...+36 is equal to the average (explained above) times the number of parcels (36), due to the homogeneity nature of the average.


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: What is the greatest prime factor of 1+2+3+….+36?  [#permalink]

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New post 08 Oct 2018, 06:37
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the greatest prime factor of \(1+2+3+….+36\)?

\(A. 2\)
\(B. 3\)
\(C. 9\)
\(D. 31\)
\(E. 37\)


Useful formula: 1 + 2 + 3 + 4 + . . . . + n = (n)(n + 1)/2

So, 1 + 2 + 3 + …. + 36 = (36)(36 + 1)/2
= (36)(37)/2
= (18)(37)
= (2)(3)(3)(17)

So, the greatest prime factor is 37

Answer: E

Cheers,
Brent
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Re: What is the greatest prime factor of 1+2+3+….+36?  [#permalink]

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New post 08 Oct 2018, 07:07
Formula: 1st term + Last term/2 x number of terms in the series

Let's apply it
=>(1+36)/2 x 36
=>37 x 18 = 666

From the above, we can arrive at the answer.

Greatest prime factor is 37 (E)
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Re: What is the greatest prime factor of 1+2+3+….+36?  [#permalink]

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New post 10 Oct 2018, 00:59
=>

Since \(1 + 2 + 3 + … + n = \frac{n(n+1)}{2}\), we have \(1 + 2 + 3 + … + 36 = \frac{(36*37)}{2} = 18*37 = 2*32*37.\)
Thus, \(37\) is the greatest prime factor of \(2*3^2*37.\)

Therefore, the answer is E.
Answer: E
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Re: What is the greatest prime factor of 1+2+3+….+36?   [#permalink] 10 Oct 2018, 00:59
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