Question : What is the sum of all three-digit numbers which are divisible by 15?
Option 1: 41200
Option 2: 36825
Option 3: 32850
Option 4: 28750
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Correct Answer: 32850
Solution :
We need to find the sum of all three-digit numbers divisible by 15.
The sum of an arithmetic series: S = $(\frac{n}{2}) \times (\text{first term} + \text{last term})$
The smallest three-digit number divisible by 15 is 105.
The largest three-digit number divisible by 15 is 990.
Using the formula to find the number of terms:
Number of terms = ($\frac{(\text{last term} – \text{first term})}{\text{common difference}}) + 1$
⇒ Number of terms = $(\frac{(990 – 105)}{15}) + 1$
⇒ Number of terms = $(\frac{885}{15}) + 1$
⇒ Number of terms = 59 + 1
$\therefore$ Number of terms = 60
Using the formula for the sum of an arithmetic series:
Sum = ($\frac{\text{number of terms}}{2}) \times (\text{first term} + \text{last term})$
⇒ Sum = ($\frac{60}{2}) \times (105 + 990)$
⇒ Sum = 30 $\times$ 1095
$\therefore$ Sum = 32850
Hence, the correct answer is 32850.
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