Let's solve question 33 step-by-step.

The question states:

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

Step 1: Express the condition mathematically

Let the number be $N$. We know: $N \equiv 3 \ (\text{mod} \ 14)$ $N \equiv 3 \ (\text{mod} \ 20)$ $N \equiv 3 \ (\text{mod} \ 35)$

This means $N$ leaves a remainder of 3 when divided by 14, 20, or 35.

Step 2: Find the Least Common Multiple (LCM)

To find the least possible value of $N$, we need to determine the least common multiple (LCM) of 14, 20, and 35. The number $N$ must satisfy the condition $N = \text{LCM}(14, 20, 35) + 3$.

First, calculate the prime factorizations:

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

The LCM takes the highest powers of all primes appearing in the factorizations:

• $2^2$ (from 20),
• $5$ (from 20 and 35),
• $7$ (from 14 and 35).

So, the LCM is: $\text{LCM}(14, 20, 35) = 2^2 \times 5 \times 7 = 140$

Step 3: Add the remainder

Since the remainder is 3, the least number $N$ is: $N = 140 + 3 = 143$

Thus, the least possible value of $N$ is 143.

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

Here are five questions that can help expand your understanding:

1. How do we determine the LCM of two or more numbers?
2. What is the significance of the modulus in congruences?
3. Why do we add the remainder after finding the LCM?
4. How can we verify if 143 meets all conditions of the problem?
5. Can this approach be applied to other remainder problems?

Tip: Always check if the solution satisfies the conditions by dividing the result by each number to verify the remainder!