# Pre-RMO 2014/1

A natural number $$k$$ is such that $$k^2 < 2014 < (k + 1)^2.$$ What is the largest prime factor of $$k$$?

This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
3 years, 11 months ago

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$$44^2 < 2014 < (44+1)^2$$. This satisfies the above inequality. Therefore, $$k = 44$$. The largest prime factor of 44 is 11. 11 is the answer.

- 3 years, 11 months ago

To find Largest prime factor of any number K , you need to do its prime factorisation. for e.g.

prime factorisation of 10 is 10=5*2

so the largest prime factor is 10

But the question you asked is something like K^2<2014<(K+1)^2 if we take K^2<2014

we get k<+-44.88….. so largest value of K is 44

we take largest value so we can get largest prime factor. prime factorisation of 44 is -

44= 2211

therefore largest prime factor of K is 11.

- 1 year, 3 months ago

11

- 3 years ago

11

- 3 years, 1 month ago

Easy....11

- 3 years, 1 month ago

11

- 3 years, 3 months ago

11

- 3 years, 10 months ago

$${44}^{2}$$=1936<2014<$${45}^{2}$$=2025,so if k <44 $$({k}+{1})^{2}$$<2014 therefore k=44 so the largest prime factor of k is 11.

- 3 years, 10 months ago

- 3 years, 11 months ago

11

- 3 years, 11 months ago

k2<2o14 ..k<√2014 ...k<44.8 ..also ...2014<k2+2k+1 ...2014,(k+1)2 ...√2014<(k+1) ...44.8<(k+1) ...43.8<k ...k=44

- 3 years, 11 months ago

11

- 3 years, 11 months ago

11

- 3 years, 11 months ago

11

- 3 years, 11 months ago

Here k=44 satisfies the inequality.Now the prime factors of k are 2 and 11, of which 11 is the largest prime factor.

- 3 years, 11 months ago

11

- 3 years, 11 months ago

11

- 3 years, 11 months ago

2014 lies between 44 and 45 square. But 45 square is 2025. Therefore we take 44. The prime factors of 44 are 2 and 11. The largest prime factor is 11 and smallest prime factor is 2.

- 3 years, 11 months ago

k^2<2014. Max. value integer value of 'k' is 44 & it also satisfy the second equation i.e. (k+1)^2 >2014 . Hence largest value of k is 44 & its biggest prime factor is 11. So answer is 11.

- 3 years, 11 months ago

<=> 0 < 2014 - k^2 < 2k +1 taking the right side of the inequallity and solving with the restrictions we get that the only integer is 44=4*11 therefore the answer is 11

- 3 years, 11 months ago

11

- 3 years, 11 months ago

11 is the answer as the valuebof k is 44. Prime factorization of 44= 2 \times 2 \times 11

- 3 years, 11 months ago