Answer:
18 numbers between 650 and 780 have remainder 4 when
divided by 7
12 numbers between 650 and 780 have remainder 6 when
divided by 11
Step-by-step explanation:
* Lets explain how to solve the problem
- The first number divisible by 7 between 650 and 780 is 651
∴ The first number between 650 and 780 has reminder 4 when
divided by 7 is 651 + 4 = 655
- The last number divisible by 7 between 650 and 780 is 777
- But we add 777 by 4 the answer will be 781 out the range of the
numbers, so will take the number just before 777 and divisible by 7,
the number is 770
∴ The last number between 650 and 780 has reminder 4 when
divided by 7 is 770 + 4 = 774
- Now lets use the rule of the arithmetic series
∵ [tex]a_{n}=a_{1}+(n-1)d[/tex], where [tex]a_{1}[/tex] is the 1st term, d is
the constant difference between each two consecutive terms and
n is the position of the term in the series
∵ [tex]a_{1}[/tex] = 655
∵ [tex]a_{n}[/tex] = 774
∵ d = 7
∴ 774 = 655 + (n - 1)(7)
- Subtract 774 from both sides
∴ 119 = (n - 1)(7)
- Divide both sides by 7
∴ 17 = n - 1
- Add both sides by 1
∴ n = 18
18 numbers between 650 and 780 have remainder 4 when
divided by 7
- By checking the number 655 when divided by 11 give remainder 6
∴ The first number between 650 and 780 has remainder 6 when
divided by 11 is 655
- By checking the number 744 when divided by 11 give remainder 4,
we need remainder 6 so add 2 to 774 to get 776
∴ The last number between 650 and 780 has reminder 6 when
divided by 11 is 776
- By using the same rule rule of the arithmetic series
∵ [tex]a_{1}[/tex] = 655
∵ [tex]a_{n}[/tex] = 776
∵ d = 11
∴ 776 = 655 + (n - 1)(11)
- Subtract 774 from both sides
∴ 121 = (n - 1)(7)
- Divide both sides by 11
∴ 11 = n - 1
- Add both sides by 1
∴ n = 12
12 numbers between 650 and 780 have remainder 6 when
divided by 11
