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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?

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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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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.

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Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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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

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=>

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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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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