Question
Download Solution PDFThe sum of all two-digit numbers which leave the remainder of 5 when they are divided by 7 is equal to
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
Two-digit numbers leaving remainder 5 when divided by 7.
Formula used:
Arithmetic Progression Sum: S = n/2 × (a + l)
where n = number of terms, a = first term, l = last term
Calculation:
Smallest two-digit number leaving remainder 5 when divided by 7:
⇒ 7 × 1 + 5 = 12
Largest two-digit number leaving remainder 5 when divided by 7:
⇒ 7 × 13 + 5 = 96
The numbers form an arithmetic progression: 12, 19, 26, ..., 96
Common difference (d) = 7
To find the number of terms (n):
⇒ l = a + (n - 1) × d
⇒ 96 = 12 + (n - 1) × 7
⇒ 84 = (n - 1) × 7
⇒ 12 = n - 1
⇒ n = 13
Sum of the numbers (S):
⇒ S = n/2 × (a + l)
⇒ S = 13/2 × (12 + 96)
⇒ S = 13 × 54
⇒ S = 702
∴ The sum is 702.
Last updated on Feb 11, 2025
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