Let's solve Question 33 step by step.

Problem:

You need to find the least possible value of a number that, when divided by 14, 20, or 35, always leaves a remainder of 3.

Step-by-Step Solution:

1. Interpretation: If a number $N$ leaves a remainder of 3 when divided by 14, 20, or 35, this means: $N \equiv 3 \pmod{14}$ $N \equiv 3 \pmod{20}$ $N \equiv 3 \pmod{35}$ Essentially, $N - 3$ must be divisible by 14, 20, and 35.

2. Find the Least Common Multiple (LCM): To find the least number that satisfies these conditions, calculate the LCM of 14, 20, and 35.

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

   The LCM is the product of the highest powers of all prime factors: $\text{LCM} = 2^2 \times 5 \times 7 = 140$

3. Find the Least Value of the Number: The number $N - 3$ must be a multiple of the LCM, so: $N - 3 = 140k \quad \text{for some integer } k.$ To get the smallest value of $N$, choose $k = 1$: $N - 3 = 140 \quad \Rightarrow \quad N = 140 + 3 = 143.$

Answer:

The least possible value of the number is 143.

Would you like any more details on this solution or have further questions?

Here are five related questions to deepen your understanding:

1. How would the solution change if the remainder was 5 instead of 3?
2. What is the prime factorization method, and how is it used to find the LCM?
3. How can you use modular arithmetic to solve similar problems?
4. How does finding the greatest common divisor (GCD) relate to finding the LCM?
5. What if the numbers were larger? How would you approach the solution?

Tip:

When calculating the LCM of multiple numbers, always take the highest power of each prime factor involved.