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Cube of Positive Integer | Number Theory | AIME I, 2015 Question 3

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2010 based on Cube of Positive Integer.

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Cube of Positive Integer.

Cube of Positive Numbers – AIME I, 2015


There is a prime number p such that 12p+1 is the cube of positive integer.Find p..

  • is 107
  • is 183
  • is 840
  • cannot be determined from the given information

Key Concepts


Algebra

Theory of Equations

Number Theory

Check the Answer


Answer: is 183.

AIME, 2015, Question 3

Elementary Number Theory by David Burton

Try with Hints


\(a^{3}=12p+1\) implies that \(a^{3}-1=12p\) that is (a-1)(\(a^{2}\)+a+1)=12p

a is odd, a-1 even, \(a^{2} +a+1 odd implies a-1 multiple of 12 that is here =12 then a=12+1 =13

\(a^{2}+a+1=p implies p= 169+13+1=183.

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