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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, 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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