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# LCM and Integers | AIME I, 1998 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 1998, Problem 1, based on LCM and Integers.

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

Lcm

Algebra

Integers

But try the problem first…

Source

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

$$6^{6}=2^{6}3^{6}$$

$$8^{8}=2^{24}$$

$$12^{12}=2^{24}3^{12}$$

Second Hint

lcm$$(6^{6},8^{8})$$=$$2^{24}3^{6}$$

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

=$$(2^{24}3^{6},2^{a}3^{b})$$

=$$2^{max(24,a)}3^{max(6,b)}$$

Final Step

$$\Rightarrow b=12, 0 \leq a \leq 24$$

$$\Rightarrow$$ number of values of k=25.

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