A government department has 19 151 employees. Is it possible to evenly divide the employees into groups of 11? Use divisibility rules to respond.
To determine if 19,151 is divisible by 11, we can use the alternating digit sum rule: Add the first digit (1), subtract the next digit (9), add the next digit (1), subtract the next digit (5), and so on. If the result is either 0 or a multiple of 11, then the original number is divisible by 11.
1 - 9 + 1 - 5 + 1 = -11
Since -11 is a multiple of 11, we can conclude that 19,151 is divisible by 11. Therefore, it is possible to evenly divide the employees into groups of 11.
To find out how many groups of 11 can be formed:
19,151 ÷ 11 = 1,741 with a remainder of 0
So there can be 1,741 groups of 11 employees formed, with no employees left over.
To determine if it is possible to evenly divide 19,151 employees into groups of 11, we can use the divisibility rule for 11.
The divisibility rule for 11 states that a number is divisible by 11 if the difference between the sum of its even-placed digits and the sum of its odd-placed digits is either 0 or a multiple of 11.
Let's calculate the sums:
Odd-placed digits: 1+1+5 = 7
Even-placed digits: 9+1 = 10
The difference between the sums is 10 - 7 = 3, which means that 19,151 is not divisible by 11.
Therefore, it is not possible to evenly divide the 19,151 employees into groups of 11.
To determine if it is possible to evenly divide the employees into groups of 11, we need to check if the total number of employees is divisible by 11. Let's follow the divisibility rule for 11.
The divisibility rule for 11 states that if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is divisible by 11, then the original number is divisible by 11.
In this case, the original number is 19,151. If we separate the digits into odd and even positions, we have:
Odd positions: 1, 1, 5
Even positions: 9, 1
The sum of the digits in the odd positions is 1 + 1 + 5 = 7.
The sum of the digits in the even positions is 9 + 1 = 10.
Now, we calculate the difference between the sums: 10 - 7 = 3.
Since the difference of 3 is not divisible by 11, we can conclude that the total number of employees, 19,151, is NOT divisible evenly into groups of 11.