Which Of The Following Numbers Are Multiples Of 9

Which of the following numbers aremultiples of 9? This question appears frequently in elementary arithmetic, competitive exams, and everyday problem‑solving situations. Understanding how to identify multiples of 9 quickly not only saves time but also builds a solid foundation for more advanced topics like factors, least common multiples, and modular arithmetic. In this guide we will explore the concept of multiples of 9, explain the reliable divisibility rule based on digit sums, walk through step‑by‑step examples, highlight common mistakes, and provide a practice set so you can test your skill immediately.

Understanding Multiples of 9

Definition

A multiple of 9 is any integer that can be expressed as (9 \times k), where (k) is an integer (positive, negative, or zero). In other words, if you can divide a number by 9 and obtain a whole number without remainder, that number is a multiple of 9.

Examples:

• (9 \times 1 = 9) → 9 is a multiple of 9.
• (9 \times 4 = 36) → 36 is a multiple of 9.
• (9 \times (-3) = -27) → –27 is also a multiple of 9 (negative multiples count as well).

The Divisibility Rule for 9

Rather than performing long division each time, mathematicians use a simple shortcut: a number is divisible by 9 if and only if the sum of its digits is divisible by 9. This rule works because 9 is one less than the base‑10 system’s radix (10), causing a neat cancellation in the positional value expansion.

Why it works (briefly):
Write a number (N) as (a_n10^n + a_{n-1}10^{n-1} + \dots + a_110 + a_0). Since (10 \equiv 1 \pmod{9}), each power of 10 is congruent to 1 modulo 9. Therefore [ N \equiv a_n + a_{n-1} + \dots + a_1 + a_0 \pmod{9}. ]
If the digit sum is a multiple of 9, the original number is too.

Applying the Rule: Step‑by‑Step Examples

Example 1: Small Numbers

Is 81 a multiple of 9?

1. Sum the digits: (8 + 1 = 9). 2. Check if the sum (9) is divisible by 9 → yes.
2. Conclusion: 81 is a multiple of 9 (indeed, (9 \times 9 = 81)).

Example 2: A Three‑Digit Number

Is 573 a multiple of 9?

1. Digit sum: (5 + 7 + 3 = 15).
2. 15 is not divisible by 9 (the nearest multiples are 9 and 18).
3. Therefore, 573 is not a multiple of 9.
   (You can verify: (9 \times 63 = 567) and (9 \times 64 = 576); 573 lies between them.)

Example 3: A Larger Number with Carrying

Is 4,896 a multiple of 9?

1. Add digits: (4 + 8 + 9 + 6 = 27).
2. 27 is divisible by 9 ((9 \times 3 = 27)).
3. Hence, 4,896 is a multiple of 9.
   (Dividing gives (4,896 ÷ 9 = 544) exactly.)

Example 4: Handling Zeros

Is 9,090 a multiple of 9?

1. Digit sum: (9 + 0 + 9 + 0 = 18).
2. 18 ÷ 9 = 2 → divisible.
3. 9,090 is a multiple of 9 ((9 \times 1,010 = 9,090)).

Notice that zeros do not affect the sum; they can be ignored safely when applying the rule.

Common Pitfalls and Misconceptions

Practice Set: Which of the Following Numbers Are Multiples of 9?

Below is a mixed list of