Understanding Divisibility by 9

The question asks for the value of the digit 'x' in the number x4562, given that the entire number is divisible by 9. To solve this, we need to use the divisibility rule for 9.

Divisibility Rule for 9 Explained

A key concept in number theory, the divisibility rule for 9 states that a number is divisible by 9 if and only if the sum of its digits is divisible by 9.

For example:

• The number 18 is divisible by 9 because the sum of its digits (1 + 8 = 9) is divisible by 9.
• The number 273 is divisible by 9 because the sum of its digits (2 + 7 + 3 = 12) is not divisible by 9. (Correction: 2+7+3=12, 12 is not divisible by 9. So 273 is not divisible by 9. Let's use a correct example: 162 -> 1+6+2=9. 162 is divisible by 9.) Let's use a correct example.
• The number 162 is divisible by 9 because the sum of its digits (1 + 6 + 2 = 9) is divisible by 9.
• The number 459 is divisible by 9 because the sum of its digits (4 + 5 + 9 = 18) is divisible by 9.

Applying the Divisibility Rule to x4562

The given number is x4562. The digits of this number are x, 4, 5, 6, and 2.

According to the divisibility rule of 9, the sum of these digits must be divisible by 9 for the number x4562 to be divisible by 9.

Let's find the sum of the digits:

Sum of digits = x + 4 + 5 + 6 + 2

Sum of digits = x + (4 + 5 + 6 + 2)

Sum of digits = x + 17

Finding the Value of x

For the number x4562 to be divisible by 9, the sum of its digits, which is $x + 17$, must be divisible by 9.

Since 'x' is a single digit and it is the first digit of a five-digit number, 'x' must be a non-zero digit, i.e., $x \in \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$.

We need to find a value of x from this set such that $x + 17$ is divisible by 9.

Let's test the possible values for x:

• If x = 1, Sum = $1 + 17 = 18$. Is 18 divisible by 9? Yes, $18 \div 9 = 2$.
• If x = 2, Sum = $2 + 17 = 19$. Is 19 divisible by 9? No.
• If x = 3, Sum = $3 + 17 = 20$. Is 20 divisible by 9? No.
• If x = 4, Sum = $4 + 17 = 21$. Is 21 divisible by 9? No.
• If x = 5, Sum = $5 + 17 = 22$. Is 22 divisible by 9? No.
• If x = 6, Sum = $6 + 17 = 23$. Is 23 divisible by 9? No.
• If x = 7, Sum = $7 + 17 = 24$. Is 24 divisible by 9? No.
• If x = 8, Sum = $8 + 17 = 25$. Is 25 divisible by 9? No.
• If x = 9, Sum = $9 + 17 = 26$. Is 26 divisible by 9? No.

The only value of x from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ for which $x + 17$ is divisible by 9 is when $x = 1$. In this case, the sum is 18, which is divisible by 9.

Therefore, the value of x must be 1.

The number is 14562. Let's check: $1 + 4 + 5 + 6 + 2 = 18$. $18 \div 9 = 2$. So, 14562 is indeed divisible by 9.

Conclusion on the Value of x

Based on the divisibility rule for 9, the sum of the digits of x4562 must be divisible by 9. By calculating the sum as $x + 17$ and checking possible digit values for x, we found that only x = 1 results in a sum (18) that is divisible by 9. Thus, the value of x is 1.

Revision Table: Divisibility Rules

Additional Information: Number Properties

Understanding divisibility rules is a fundamental part of number theory and is essential for solving problems involving factors, multiples, and prime numbers. The divisibility rule for 9 is closely related to the rule for 3, as 9 is $3^2$. Both rules rely on the sum of the digits.

• Place Value: In the number x4562, the digit x is in the ten thousands place, 4 is in the thousands place, 5 in the hundreds place, 6 in the tens place, and 2 in the units place. The number can be written as $10000x + 4000 + 500 + 60 + 2$.
• Why the Sum of Digits Rule Works for 9: Any number can be expressed as a sum of multiples of powers of 10. For example, $456 = 4 \times 100 + 5 \times 10 + 6 \times 1$. Since $10^n - 1$ is always a sequence of n nines (e.g., $10-1=9$, $100-1=99$, $1000-1=999$), $10^n$ is always $1$ more than a multiple of 9 ($10^n = 9k + 1$ for some integer k). So, $4 \times 100 + 5 \times 10 + 6 \times 1 = 4 \times (99+1) + 5 \times (9+1) + 6 \times 1 = (4 \times 99 + 4) + (5 \times 9 + 5) + 6 = (4 \times 99 + 5 \times 9) + (4+5+6)$. The first part $(4 \times 99 + 5 \times 9)$ is clearly a multiple of 9. Therefore, the divisibility of the entire number by 9 depends only on whether the second part, the sum of the digits $(4+5+6)$, is divisible by 9. This principle extends to any number of digits, including x4562.