Number Relationship Puzzle Explained

This type of question requires identifying a logical or arithmetic pattern between two numbers in the first pair and then applying that exact same pattern to the second pair to find the missing number.

We are presented with the relationship: \(51728 : 73940\) and need to find the missing number for \(38641 : ?\).

Identifying the Arithmetic Pattern

Let's analyze the relationship between the digits of \(51728\) and \(73940\). We compare each digit position:

• Position 1: \(5 \rightarrow 7\). The difference is \(7 - 5 = 2\).
• Position 2: \(1 \rightarrow 3\). The difference is \(3 - 1 = 2\).
• Position 3: \(7 \rightarrow 9\). The difference is \(9 - 7 = 2\).
• Position 4: \(2 \rightarrow 4\). The difference is \(4 - 2 = 2\).
• Position 5: \(8 \rightarrow 0\). This doesn't seem like a simple addition of \(2\). Let's consider adding \(2\) and taking the result modulo \(10\).

Checking the modulo \(10\) idea:

• \(5 + 2 = 7\)
• \(1 + 2 = 3\)
• \(7 + 2 = 9\)
• \(2 + 2 = 4\)
• \(8 + 2 = 10\). Applying modulo \(10\): \(10 \pmod{10} = 0\).

The pattern is confirmed: Add \(2\) to each digit. If the sum is \(10\) or greater, use the remainder when divided by \(10\) (i.e., the last digit). This operation can be represented as adding \(2\) modulo \(10\) to each digit.

Applying the Pattern to Find the Missing Number

Now, we apply this established pattern (add \(2\) to each digit, modulo \(10\)) to the number \(38641\):

• First digit: \(3 + 2 = 5\)
• Second digit: \(8 + 2 = 10\). Modulo \(10\) gives \(0\).
• Third digit: \(6 + 2 = 8\)
• Fourth digit: \(4 + 2 = 6\)
• Fifth digit: \(1 + 2 = 3\)

Putting these digits together, the resulting number is \(50863\).

Thus, the missing number in the sequence \(38641 : ?\) is \(50863\).