Let's solve the problems shown in the image:

Question 33:

"When a number is divided by 14, 20, or 35, the remainder is always 3. Find the least possible value of the number."

We are asked to find a number $N$ such that:

• $N \equiv 3 \pmod{14}$
• $N \equiv 3 \pmod{20}$
• $N \equiv 3 \pmod{35}$

This is a system of simultaneous congruences, and the solution is to find the least number $N$ that satisfies:

$N = LCM(14, 20, 35) + 3$

First, let's find the LCM of 14, 20, and 35:

• Prime factors:
  • $14 = 2 \times 7$
  • $20 = 2^2 \times 5$
  • $35 = 5 \times 7$

The LCM is the product of the highest powers of all prime factors involved: $\text{LCM}(14, 20, 35) = 2^2 \times 5 \times 7 = 140.$

Therefore, the least possible value of the number $N$ is: $N = 140 + 3 = 143.$

Question 34:

Tommy has 102 blue pens, 144 red pens, and 210 green pens. He packs all the pens of the same color in packets such that the numbers of pens in all packets are the same.

Part (a):

At most how many pens are there in each packet?

We are asked to find the greatest number of pens per packet, which corresponds to the greatest common divisor (GCD) of 102, 144, and 210.

• Prime factorization:
  • $102 = 2 \times 3 \times 17$
  • $144 = 2^4 \times 3^2$
  • $210 = 2 \times 3 \times 5 \times 7$

The common factors are $2 \times 3 = 6$.

Thus, the greatest number of pens per packet is 6.

Part (b):

If the number of pens in each packet is equal to the number found in (a), how many packets of pens are there?

We need to divide the total number of pens of each color by 6:

• For blue pens: $\frac{102}{6} = 17$ packets.
• For red pens: $\frac{144}{6} = 24$ packets.
• For green pens: $\frac{210}{6} = 35$ packets.

Thus, the total number of packets is: $17 + 24 + 35 = 76 \text{ packets}.$

Would you like further details on any of the steps? Here are 5 related questions to expand on these concepts:

1. How is the least common multiple (LCM) used to solve congruence problems?
2. What are the applications of the greatest common divisor (GCD) in dividing objects equally?
3. How would the answer change if the remainder in Question 33 was different?
4. Can you explain the prime factorization method for finding LCM and GCD?
5. How does modular arithmetic simplify problems involving remainders?

Tip:

When solving systems of congruences, using the least common multiple (LCM) helps you find common solutions efficiently!