Draw circles around the multiples of 6. Draw triangles around the multiples of 9. Which numbers have a circle and a triangle?

Here's how to solve the problem by identifying multiples:

Step 1: Identify Multiples of 6 A number is a multiple of 6 if it is divisible by both 2 (even) and 3 (sum of digits is a multiple of 3).

• 126 (even, $1+2+6=9$, divisible by 3)
• 24 (even, $2+4=6$, divisible by 3)
• 72 (even, $7+2=9$, divisible by 3)
• 66 (even, $6+6=12$, divisible by 3)
• 108 (even, $1+0+8=9$, divisible by 3)
• 414 (even, $4+1+4=9$, divisible by 3)
• 42 (even, $4+2=6$, divisible by 3)
• 30 (even, $3+0=3$, divisible by 3)
• 54 (even, $5+4=9$, divisible by 3)
• 18 (even, $1+8=9$, divisible by 3)
• 48 (even, $4+8=12$, divisible by 3)

The multiples of 6 are: $\boxed{126, 24, 72, 66, 108, 414, 42, 30, 54, 18, 48}$. (You would draw circles around these numbers on your paper.)

Step 2: Identify Multiples of 9 A number is a multiple of 9 if the sum of its digits is a multiple of 9.

• 126 ($1+2+6=9$, divisible by 9)
• 27 ($2+7=9$, divisible by 9)
• 243 ($2+4+3=9$, divisible by 9)
• 72 ($7+2=9$, divisible by 9)
• 108 ($1+0+8=9$, divisible by 9)
• 414 ($4+1+4=9$, divisible by 9)
• 81 ($8+1=9$, divisible by 9)
• 63 ($6+3=9$, divisible by 9)
• 54 ($5+4=9$, divisible by 9)
• 18 ($1+8=9$, divisible by 9)

The multiples of 9 are: $\boxed{126, 27, 243, 72, 108, 414, 81, 63, 54, 18}$. (You would draw triangles around these numbers on your paper.)

Step 3: Identify numbers with both a circle and a triangle These are the numbers that are multiples of both 6 and 9. This means they are multiples of the least common multiple (LCM) of 6 and 9, which is 18. Comparing the lists from Step 1 and Step 2, the numbers that appear in both are:

• 126
• 72
• 108
• 414
• 54
• 18

The numbers that have a circle and a triangle are: $\boxed{18, 54, 72, 108, 126, 414}$.