To check if a number is divisible by 11, we use the divisibility rule for 11. The rule states that 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.
Let's apply this rule to the given numbers:
### Part (a): Number 1048564
1. Write down the digits in their respective positions:
- Positions: 1 2 3 4 5 6 7
- Digits: 1 0 4 8 5 6 4
2. Identify the digits in the odd and even positions:
- Odd positions (1st, 3rd, 5th, 7th): 1, 4, 5, 4
- Even positions (2nd, 4th, 6th): 0, 8, 6
3. Calculate the sum of the digits in the odd positions:
- Odd positions sum: [tex]\(1 + 4 + 5 + 4 = 14\)[/tex]
4. Calculate the sum of the digits in the even positions:
- Even positions sum: [tex]\(0 + 8 + 6 = 14\)[/tex]
5. Find the difference between the sums:
- Difference: [tex]\(14 - 14 = 0\)[/tex]
Since the difference is 0, which is a multiple of 11, the number 1048564 is divisible by 11.
### Part (b): Number 1120043
1. Write down the digits in their respective positions:
- Positions: 1 2 3 4 5 6 7
- Digits: 1 1 2 0 0 4 3
2. Identify the digits in the odd and even positions:
- Odd positions (1st, 3rd, 5th, 7th): 1, 2, 0, 3
- Even positions (2nd, 4th, 6th): 1, 0, 4
3. Calculate the sum of the digits in the odd positions:
- Odd positions sum: [tex]\(1 + 2 + 0 + 3 = 6\)[/tex]
4. Calculate the sum of the digits in the even positions:
- Even positions sum: [tex]\(1 + 0 + 4 = 5\)[/tex]
5. Find the difference between the sums:
- Difference: [tex]\(6 - 5 = 1\)[/tex]
Since the difference is 1, which is not a multiple of 11, the number 1120043 is not divisible by 11.
### Final Results
a) The number 1048564 is divisible by 11.
b) The number 1120043 is not divisible by 11.
