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

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by Max@Math Revolution » Mon Oct 08, 2018 12:49 am

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

by fskilnik@GMATH » Mon Oct 08, 2018 4:38 am
Max@Math Revolution 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
$$?\,\,\,:\,\,\,{\rm{greatest}}\,\,{\rm{prime}}\,\,{\rm{factor}}$$
$$1 + 2 + 3 + \ldots + 36\,\,\mathop = \limits^{\left( * \right)} \,\,\left( {{{1 + 36} \over 2}} \right) \cdot 36 = 18 \cdot 37\,\,\,\,\mathop \Rightarrow \limits^{2,3\,\,{\rm{or}}\,\,37} \,\,\,\,? = 37$$
(*) The parcels 1, 2, 3, ..., 36 constitute 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.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Brent@GMATPrepNow » Mon Oct 08, 2018 5:34 am
Max@Math Revolution wrote: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
Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Tue Oct 09, 2018 11:49 pm
=>

Since 1 + 2 + 3 + ... + n = n(n+1)/2, we have 1 + 2 + 3 + ... + 36 = (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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by Scott@TargetTestPrep » Wed Oct 10, 2018 5:48 pm
Max@Math Revolution 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
We can find the sum of this equally-spaced set by first determining the average of the set and then multiplying the average by the number of items in the set. We calculate the average as: (smallest number + greatest number)/2. We note that there are 36 numbers in the set. Thus, we have:

(1+2+3+....+36) = (1 + 36)/2 x 36 = 37 x 18 = 37 x 2 x 3^2

So the greatest prime factor is 37.

Answer: E

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