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# What is the greatest prime factor of 25^2 + 44^2 + 2200?

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What is the greatest prime factor of 25^2 + 44^2 + 2200? [#permalink]
VeritasKarishma How can the problem above be recognized promptly? It is so tricky one. Could you please make an attempt to resolve the problem above? Thanks a ton in advance if anyone helps.
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Re: What is the greatest prime factor of 25^2 + 44^2 + 2200? [#permalink]
masuda wrote:
VeritasKarishma How can the problem above be recognized promptly? It is so tricky one. Could you please make an attempt to resolve the problem above? Thanks a ton in advance if anyone helps.

a solution to the problem above is in the link below:

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Re: What is the greatest prime factor of 25^2 + 44^2 + 2200? [#permalink]
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masuda wrote:
VeritasKarishma How can the problem above be recognized promptly? It is so tricky one. Could you please make an attempt to resolve the problem above? Thanks a ton in advance if anyone helps.

Doesn't matter if you don't recognise it instantly. To find the greatest prime factor, you need to break the whole expression down into its factors.

$$25^2 + 44^2 + 2200$$

The first obvious step would be to take common to ease the calculations. Let's see what we have common in the terms.

$$5^4 + 2^4*11^2 * 2^3*5^2*11$$

Note that the first term has 5s only, the second term has 11s and 2s and the third term has both. We can't take anything common. With the squares in the first two terms, you should think of a^2 + b^2 + 2ab. On inspection you see that the third term is 2ab.

$$= (25 + 44)^2$$

= 69^2 = (3*23)^2

It's now broken down into its factors. 23 is the greatest prime factor.
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Re: What is the greatest prime factor of 25^2 + 44^2 + 2200? [#permalink]
nick1816 Could you please explain how you have caught the tricky PS above is a reflection of a formula?
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Re: What is the greatest prime factor of 25^2 + 44^2 + 2200? [#permalink]
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Turjoy98 wrote:
nick1816 Could you please explain how you have caught the tricky PS above is a reflection of a formula?

Turjoy98

Usually a hunch plays a role in such problems. Since we have first two terms as $$25^2+44^2$$ so we try to see how much short or excess the number 2200 is than 2*25*44 which in this case, on checking, results in zero excess or short from 2200

If there was some short or excess even then we would have translated the expression in the algebraic formula to make it simpler.
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What is the greatest prime factor of 25^2 + 44^2 + 2200? [#permalink]
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Turjoy98

Nice explanation by GMATinsight. My explanation is bit similar.

Most probably you get $$(a+b)^2$$ or $$(a+b)^2-c^2$$ = (a+b+c)(a+b-c) in such questions.

Or there is a direct observation- 2200= 22*100 = 22*(2*2*25) = 44*2*25

Turjoy98 wrote:
nick1816 Could you please explain how you have caught the tricky PS above is a reflection of a formula?
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Re: What is the greatest prime factor of 25^2 + 44^2 + 2200? [#permalink]
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Re: What is the greatest prime factor of 25^2 + 44^2 + 2200? [#permalink]
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