Let the number be $N$. The given conditions can be translated into the following modular equations:

1. When divided by 4, the remainder is 1: $N \equiv 1 \pmod{4}$

2. When divided by 5, the remainder is 4: $N \equiv 4 \pmod{5}$

These two conditions imply the number $N$ must satisfy both these congruences. We can solve this system using the Chinese Remainder Theorem or a straightforward method of successive substitution.

Step 1: Solve the system of congruences

We need to find an integer $N$ that satisfies both: $N \equiv 1 \pmod{4} \quad \text{and} \quad N \equiv 4 \pmod{5}$

Start by expressing $N$ in terms of one of the congruences. From the first congruence, we can write: $N = 4k + 1 \quad \text{for some integer } k.$ Now substitute this into the second congruence: $4k + 1 \equiv 4 \pmod{5}.$ Simplifying this: $4k \equiv 3 \pmod{5}.$ Now, multiply both sides of the equation by the modular inverse of 4 modulo 5. The inverse of 4 modulo 5 is 4, since $4 \times 4 = 16 \equiv 1 \pmod{5}$. Multiply both sides by 4: $16k \equiv 12 \pmod{5}.$ Simplify: $k \equiv 2 \pmod{5}.$ Thus, $k = 5m + 2$ for some integer $m$.

Step 2: Substitute back to find $N$

Now substitute $k = 5m + 2$ back into the expression for $N$: $N = 4k + 1 = 4(5m + 2) + 1 = 20m + 9.$ Thus, $N = 20m + 9$ is the general solution. The smallest $N$ occurs when $m = 0$, which gives: $N = 9.$

Step 3: Find the remainders when divided by 5 and 4

Now we are asked to find the remainders when this number $N = 9$ is divided successively by 5 and 4:

1. Dividing $N = 9$ by 5: $9 \div 5 = 1 \text{ remainder } 4.$ So the remainder when divided by 5 is 4.

2. Dividing $N = 9$ by 4: $9 \div 4 = 2 \text{ remainder } 1.$ So the remainder when divided by 4 is 1.

Thus, when the number is divided successively by 5 and 4, the remainders are:

• Remainder when divided by 5: 4
• Remainder when divided by 4: 1

To Summarize:

• When $N = 9$, the remainder when divided by 5 is 4, and the remainder when divided by 4 is 1.

Would you like more details or have any further questions?

Related Questions:

1. How can we solve a system of congruences using the Chinese Remainder Theorem?
2. What is the inverse of a number modulo another number, and how is it computed?
3. How do you approach problems where you need to find remainders from multiple divisions?
4. Can we extend this problem to more complex systems involving more divisors?
5. What is the general method for solving a system of linear congruences?

Tip: When solving modular equations, always look for opportunities to simplify by isolating terms or using known identities like the modular inverse to make your work easier.