At a party, every 4 th guest recelves a key chain and every 9 th guest receives a mug. What is the position of the first guest who will receive both a key chain and a mug?

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Let the position of the guest who receives both items be $$ n $$.

Since every 4th guest gets a key chain, $$ n $$ must be a multiple of $$ 4 $$.

Since every 9th guest gets a mug, $$ n $$ must be a multiple of $$ 9 $$.

Thus, $$ n $$ must be a common multiple of $$ 4 $$ and $$ 9 $$. We are looking for the smallest such number, which is the least common multiple (LCM) of $$ 4 $$ and $$ 9 $$.

To compute the LCM, note that:

$$
\text{gcd}(4, 9) = 1
$$

Then,

$$
\text{lcm}(4, 9) = \frac{4 \times 9}{\text{gcd}(4,9)} = \frac{36}{1} = 36
$$

Therefore, the first guest to receive both a key chain and a mug is in position $$ 36 $$.
The first guest to receive both a key chain and a mug is in position 36.

At a party, every 4 th guest recelves a key chain and every 9 th guest receives a mug. What is the position of the first guest who will receive both a key chain and a mug?

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