Complete the Number 427?: Finding the Missing Digit for Divisibility by 9

Complete the number so that it is divisible by 9 without a remainder:

$427?$

The problem asks us to find the digit represented by '?' in the number $427?$ so that the entire number is divisible by 9.

To solve this problem, we will use the divisibility rule for 9, which states: A number is divisible by 9 if the sum of its digits is divisible by 9.

Let's calculate the sum of the known digits:

• The digits are 4, 2, and 7.
• Calculate the sum: $4 + 2 + 7 = 13$.

Now let's include the unknown digit, represented by $x$, in the sum. The total sum of the digits will be $S = 4 + 2 + 7 + x = 13 + x$.

We need $13 + x$ to be divisible by 9. So, find a value for $x$ such that $13 + x \equiv 0 \pmod{9}$.

Let's check each value for $x$ from 0 to 9:

• If $x = 0$, $13 + 0 = 13$, not divisible by 9.
• If $x = 1$, $13 + 1 = 14$, not divisible by 9.
• If $x = 2$, $13 + 2 = 15$, not divisible by 9.
• If $x = 3$, $13 + 3 = 16$, not divisible by 9.
• If $x = 4$, $13 + 4 = 17$, not divisible by 9.
• If $x = 5$, $13 + 5 = 18$, which is divisible by 9.

Thus, the sum of the digits is divisible by 9 when $x = 5$.

The correct digit to complete the number such that it is divisible by 9 is $5$.