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

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

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TopNewest\(44^2 < 2014 < (44+1)^2\). This satisfies the above inequality. Therefore, \(k = 44\). The largest prime factor of 44 is 11. 11 is the answer. – Sharky Kesa · 2 years, 7 months ago

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11 – Anunoy Chakraborty · 1 year, 8 months ago

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11 – Gaurav Singh · 1 year, 9 months ago

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Easy....11 – Akshat Sharda · 1 year, 9 months ago

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11 – Mridul Jain · 1 year, 11 months ago

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11 – Asish Sharma · 2 years, 6 months ago

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\({44}^{2}\)=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. – Kristian Vasilev · 2 years, 6 months ago

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11 is the answer – Vaibhav Sharma · 2 years, 7 months ago

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11 – Vikram Bishla · 2 years, 7 months ago

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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 – Wafaa Abd El Aziz · 2 years, 7 months ago

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11 – Vishal Rai · 2 years, 7 months ago

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11 – Zahra Y · 2 years, 7 months ago

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11 – Palash Som · 2 years, 7 months ago

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Here k=44 satisfies the inequality.Now the prime factors of k are 2 and 11, of which 11 is the largest prime factor. – Vivek Rao · 2 years, 7 months ago

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11 – Shudipta _Cuet12 · 2 years, 7 months ago

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11 – Subhajit Ghosh · 2 years, 7 months ago

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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. – Harish Krishnan · 2 years, 7 months ago

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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. – Rahul Verma · 2 years, 7 months ago

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<=> 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 – Carlos David Nexans · 2 years, 7 months ago

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11 – Pooja Deshmukh · 2 years, 7 months ago

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11 is the answer as the valuebof k is 44. Prime factorization of 44= 2 \times 2 \times 11 – Pulkit Kapoor · 2 years, 7 months ago

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