The solution shown in the image contains an incorrect approach to solving the problem. Let's go through the correct method.

Given:

• 90 balloons
• 60 pencils
• 150 toys

The goal is to find the largest number of identical birthday packs that can be made, meaning we need to find the greatest common divisor (GCD) of these three numbers.

Step 1: Find the GCD of 90, 60, and 150

Prime factorizations:

• $90 = 2 \times 3^2 \times 5$
• $60 = 2^2 \times 3 \times 5$
• $150 = 2 \times 3 \times 5^2$

The common factors are 2, 3, and 5, and their lowest powers are: $GCD = 2^1 \times 3^1 \times 5^1 = 30$

Thus, the largest number of birthday packs that can be made is 30 (not 13 as written in the image).

Step 2: Find the number of pencils in each pack

Each pack will have: $\frac{60}{30} = 2 \text{ pencils}$

This part is correct in the image.

Final Answers:

(a) 30 packs
(b) 2 pencils in each pack

Would you like a further breakdown or explanation? Here are some related questions:

1. How do you find the GCD of three numbers using the Euclidean algorithm?
2. What is the difference between GCD and LCM (Least Common Multiple)?
3. How would the answer change if the numbers were different?
4. Can you use the prime factorization method for larger numbers efficiently?
5. What are real-life applications of GCD in packing and distribution?

Tip: The GCD helps in splitting items into the largest equal groups, while LCM helps in finding the smallest common multiple.