Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 1998 based on LCM and Integers.

Lcm and Integer – AIME I, 1998

Find the number of values of k in \(12^{12}\) the lcm of the positive integers \(6^{6}\), \(8^{8}\) and k.

  • is 107
  • is 25
  • is 840
  • cannot be determined from the given information

Key Concepts




Check the Answer

But try the problem first…

Answer: is 25.

Suggested Reading

AIME I, 1998, Question 1

Elementary Number Theory by Sierpinsky

Try with Hints

First hint

here \(k=2^{a}3^{b}\) for integers a and b




Second Hint


\(12^{12}=2^{24}3^{12}\)=lcm of \((6^{6},8^{6})\) and k



Final Step

\(\Rightarrow b=12, 0 \leq a \leq 24\)

\(\Rightarrow\) number of values of k=25.

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