Check the divisibility of the following numbers by 9: 1) 108 2) 616 3) 294 4) 432 5) 927 Please tell me if each number is divisible by 9.
The Correct Answer and Explanation is:
To check if a number is divisible by 9, we use a simple rule: Add all the digits of the number. If the sum is divisible by 9, then the number is also divisible by 9.
Let us apply this rule to each number:
- 108
Digits: 1 + 0 + 8 = 9
9 is divisible by 9, so 108 is divisible by 9 - 616
Digits: 6 + 1 + 6 = 13
13 is not divisible by 9, so 616 is not divisible by 9 - 294
Digits: 2 + 9 + 4 = 15
15 is not divisible by 9, so 294 is not divisible by 9 - 432
Digits: 4 + 3 + 2 = 9
9 is divisible by 9, so 432 is divisible by 9 - 927
Digits: 9 + 2 + 7 = 18
18 is divisible by 9, so 927 is divisible by 9
Explanation
The divisibility rule for 9 is based on properties of the base-10 number system. When a number is written in decimal form, such as 432, it means 4 hundreds plus 3 tens plus 2 ones. Mathematically, this is 4×100 + 3×10 + 2×1. Since 10 and its powers (like 10, 100, 1000) leave a remainder of 1 when divided by 9, this means we can effectively check a number’s divisibility by simply adding its digits.
For example, the number 432 becomes 4 + 3 + 2 = 9. Since 9 is divisible by 9, the original number must also be divisible by 9. The same applies to 108 and 927. Their digits sum to 9 and 18, respectively, and both are divisible by 9.
On the other hand, numbers like 616 and 294 have digit sums of 13 and 15. Neither 13 nor 15 is divisible by 9, which means the original numbers are also not divisible by 9.
This rule is useful because it allows us to quickly test large numbers for divisibility without having to perform long division. It saves time and helps with mental math. This rule only works for 9 and similar logic applies to the rule for 3, but not for numbers like 7 or 11.
