Understanding the Problem

The question asks for the smallest 4-digit number that leaves a remainder of 3 when divided by 4, 6, 7, and 9. This means the number, let's call it N, satisfies these conditions:

• \(N \div 4\) leaves a remainder of 3
• \(N \div 6\) leaves a remainder of 3
• \(N \div 7\) leaves a remainder of 3
• \(N \div 9\) leaves a remainder of 3
• \(N\) must be the smallest possible number that is greater than or equal to 1000 (the smallest 4-digit number).

Finding the Key Concept: LCM

When a number leaves the same remainder upon division by several different numbers, it suggests using the concept of the Least Common Multiple (LCM). If we subtract the common remainder (3 in this case) from the number N, the result (\(N-3\)) must be perfectly divisible by 4, 6, 7, and 9.

Therefore, \(N-3\) must be a multiple of the LCM of 4, 6, 7, and 9. The number N can be expressed as:

\(N = \text{LCM}(4, 6, 7, 9) \times k + 3\), where \(k\) is a positive integer.

Calculating the LCM of 4, 6, 7, and 9

To find the LCM, we first find the prime factorization of each number:

• \(4 = 2^2\)
• \(6 = 2 \times 3\)
• \(7 = 7\) (7 is a prime number)
• \(9 = 3^2\)

The LCM is found by taking the highest power of each prime factor present in any of the numbers:

LCM\((4, 6, 7, 9) = 2^2 \times 3^2 \times 7^1\)

LCM\((4, 6, 7, 9) = 4 \times 9 \times 7\)

LCM\((4, 6, 7, 9) = 36 \times 7\)

LCM\((4, 6, 7, 9) = 252\)

Finding the Smallest 4-Digit Number

Now we know that the number N must be in the form \(N = 252 \times k + 3\). We need the smallest such number that is a 4-digit number, meaning \(N \ge 1000\).

We can set up the inequality:

\(252 \times k + 3 \ge 1000\)

Subtract 3 from both sides:

\(252 \times k \ge 997\)

Divide by 252:

\(k \ge \frac{997}{252}\)

\(k \ge 3.956...\)

Since \(k\) must be an integer, the smallest integer value for \(k\) that satisfies this condition is \(k = 4\).

Now, substitute \(k=4\) back into the formula for N:

\(N = 252 \times 4 + 3\)

\(N = 1008 + 3\)

\(N = 1011\)

Verifying the Result

Let's check if 1011 satisfies the conditions:

• \(1011 \div 4 = 252\) with a remainder of \(1011 - (252 \times 4) = 1011 - 1008 = 3\).
• \(1011 \div 6 = 168\) with a remainder of \(1011 - (168 \times 6) = 1011 - 1008 = 3\).
• \(1011 \div 7 = 144\) with a remainder of \(1011 - (144 \times 7) = 1011 - 1008 = 3\).
• \(1011 \div 9 = 112\) with a remainder of \(1011 - (112 \times 9) = 1011 - 1008 = 3\).

The number 1011 is indeed the smallest 4-digit number that leaves a remainder of 3 when divided by 4, 6, 7, or 9.