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Which number is divisible by both 9 and ...

Which number is divisible by both 9 and 11?

A

a)10098

B

b)10108

C

c)10089

D

d)10087

Text Solution

AI Generated Solution

The correct Answer is:
To determine which number is divisible by both 9 and 11, we can follow these steps: ### Step 1: Understand the divisibility rules for 9 and 11 - **Divisibility by 9**: A number is divisible by 9 if the sum of its digits is divisible by 9. - **Divisibility by 11**: A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or a multiple of 11. ### Step 2: Identify the number Let's consider a number, say 10089 (as mentioned in the video), and check its divisibility by both 9 and 11. ### Step 3: Check divisibility by 9 1. Calculate the sum of the digits of 10089: - \(1 + 0 + 0 + 8 + 9 = 18\) 2. Check if 18 is divisible by 9: - \(18 \div 9 = 2\) (which is an integer) - Therefore, 10089 is divisible by 9. ### Step 4: Check divisibility by 11 1. Identify the digits in odd and even positions: - Odd positions: 1 (1st), 0 (3rd), 9 (5th) → Sum = \(1 + 0 + 9 = 10\) - Even positions: 0 (2nd), 8 (4th) → Sum = \(0 + 8 = 8\) 2. Calculate the difference: - Difference = \(10 - 8 = 2\) 3. Check if 2 is divisible by 11: - Since 2 is not 0 or a multiple of 11, 10089 is not divisible by 11. ### Step 5: Conclusion Since 10089 is divisible by 9 but not by 11, we need to find another number. ### Step 6: Find the least common multiple (LCM) of 9 and 11 1. The LCM of 9 and 11 is calculated as: - \(LCM(9, 11) = 9 \times 11 = 99\) 2. Therefore, any multiple of 99 will be divisible by both 9 and 11. ### Step 7: Verify a multiple of 99 1. Let's check 99: - Sum of digits: \(9 + 9 = 18\) (divisible by 9) - Odd positions: 9 (1st), 9 (3rd) → Sum = 18 - Even positions: 9 (2nd) → Sum = 9 - Difference = \(18 - 9 = 9\) (not divisible by 11) 2. Check 198: - Sum of digits: \(1 + 9 + 8 = 18\) (divisible by 9) - Odd positions: 1 (1st), 8 (3rd) → Sum = 9 - Even positions: 9 (2nd) → Sum = 9 - Difference = \(9 - 9 = 0\) (divisible by 11) ### Final Answer The number 198 is divisible by both 9 and 11. ---
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