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March 25, 2020

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

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


First hint

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

Second Hint

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

Final Step

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

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