What is the sum of all integers between 50 and 500 which are divisible by 7?
This article will walk you through a straightforward mathematical problem: finding the sum of all integers that fall between 50 and 500 and are perfectly divisible by 7. We'll break down the process step-by-step, making it easy for anyone to understand.
Understanding the Problem
When we talk about "integers between 50 and 500," we mean all the whole numbers starting from 51 up to 499. We are not including 50 or 500 themselves. The crucial part of this problem is identifying which of these numbers can be divided by 7 with no remainder. Once we have that list, we'll add them all up to get our final answer.
Step 1: Finding the First Multiple of 7
Our first task is to find the smallest integer greater than 50 that is divisible by 7. We can do this by dividing 50 by 7:
50 ÷ 7 = 7 with a remainder of 1.
This tells us that 7 times 7 is 49, which is less than 50. The next multiple of 7 will be the first one in our range. To find it, we can take 7 times 8:
7 × 8 = 56.
So, 56 is the first integer greater than 50 that is divisible by 7.
Step 2: Finding the Last Multiple of 7
Next, we need to find the largest integer less than 500 that is divisible by 7. We'll do the same division process, but this time with 500:
500 ÷ 7 = 71 with a remainder of 3.
This means 7 times 71 is 497, which is less than 500. The remainder of 3 tells us that if we added 4 to 497, we would get 501, which is over our limit. Therefore, 497 is the last integer less than 500 that is divisible by 7.
Step 3: Identifying the Sequence of Numbers
We now have the beginning and end of our sequence of numbers divisible by 7:
- First number: 56
- Last number: 497
The numbers in this sequence will increase by 7 each time: 56, 63, 70, and so on, until we reach 497. This forms an arithmetic progression.
Step 4: Calculating the Number of Terms in the Sequence
To find the sum, we need to know how many numbers are in this sequence. We can use a formula for arithmetic progressions, or we can think of it this way:
The sequence is 7 × 8, 7 × 9, 7 × 10, ..., 7 × 71.
The number of terms is simply the count of the multipliers from 8 to 71. To find this, we subtract the smaller number from the larger number and add 1:
Number of terms = 71 - 8 + 1 = 63 + 1 = 64.
There are 64 integers between 50 and 500 that are divisible by 7.
Step 5: Calculating the Sum of the Sequence
Now that we know the first term, the last term, and the number of terms, we can use the formula for the sum of an arithmetic series. The formula is:
Sum = (Number of terms / 2) × (First term + Last term)
Plugging in our values:
Sum = (64 / 2) × (56 + 497)
Sum = 32 × (553)
Sum = 17696.
Therefore, the sum of all integers between 50 and 500 which are divisible by 7 is 17,696.
Frequently Asked Questions (FAQ)
How did you find the first number divisible by 7?
To find the first number greater than 50 that is divisible by 7, we divided 50 by 7 and found a remainder. We then identified the next multiple of 7 after 50 by either adding the difference between 7 and the remainder to 50, or by simply finding the next whole number that 7 divides into evenly, which was 56 (7 times 8).
Why did you subtract and add 1 to find the number of terms?
When counting a sequence of numbers, if you simply subtract the starting number from the ending number, you are essentially counting the "gaps" between the numbers. To include both the starting and ending numbers in your count, you need to add 1. For example, to count the numbers from 3 to 5 (3, 4, 5), you do 5 - 3 + 1 = 3.
Can this method be used for any range and any divisor?
Yes, this method is a general approach for finding the sum of any arithmetic progression. You can adapt it to find the sum of integers within any given range that are divisible by any other number by following the same steps: identify the first and last multiples, count the number of terms, and then apply the sum formula.
