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10 Aug 2020, 00:48
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D
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Difficulty:
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Question Stats:
58% (02:44) correct
42%
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The sum of all three digit numbers which leave a remainder of 3, when divided by 7 is?
A.64215
B.70720
C.64320
D.70821
E.70880
A.64215
B.70720
C.64320
D.70821
E.70880
10 Aug 2020, 03:45
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The three-digit number range is from 100 to 999 (inclusive).
The first three-digit number completely divisible by 7 is 105 and the last three-digit number completely divisible by 7 is 994.
But, as we need the remainder of 3 after division, the first three-digit number will be 101 ( when divided by 7 leaves remainder 3).
So, the first number is 101 and the last number is 997 with common difference of 7. {101,108,115,.....997}
Note: 101 = 7*14+3 and 997 = 7*142 + 3 (Don't get confuse - common difference of '7' is between numbers which are divided by 7 leaves remainder of 3).
Number of terms in an A.P. : \(T_n\) = a+(n-1)d
=> 997 = 101 + (n-1)7
=> 997-101+7=7n
=> n = 129.
Sum of an A.P. when first and last term is known: \(\frac{n}{2}\)[first term + last term]
=> \(\frac{129 }{ 2}\) [ 101+997]
=> 70,821
Answer D
The first three-digit number completely divisible by 7 is 105 and the last three-digit number completely divisible by 7 is 994.
But, as we need the remainder of 3 after division, the first three-digit number will be 101 ( when divided by 7 leaves remainder 3).
So, the first number is 101 and the last number is 997 with common difference of 7. {101,108,115,.....997}
Note: 101 = 7*14+3 and 997 = 7*142 + 3 (Don't get confuse - common difference of '7' is between numbers which are divided by 7 leaves remainder of 3).
Number of terms in an A.P. : \(T_n\) = a+(n-1)d
=> 997 = 101 + (n-1)7
=> 997-101+7=7n
=> n = 129.
Sum of an A.P. when first and last term is known: \(\frac{n}{2}\)[first term + last term]
=> \(\frac{129 }{ 2}\) [ 101+997]
=> 70,821
Answer D
10 Aug 2020, 04:21
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Hi MathRevolution
Got the same answer and used the same approach. But took some time to figure out the last value 997.
Is there a faster way to identify.
I divided 999 by 7 and back-calculated to 997.
Thanks in Advance.
Got the same answer and used the same approach. But took some time to figure out the last value 997.
Is there a faster way to identify.
I divided 999 by 7 and back-calculated to 997.
Thanks in Advance.
