Find the sum of the multiples of 11 or 5 under 551

[HellBz]

[FONT=&quot]Find the sum of the multiples of 11 or 5 under 551[/FONT]
[FONT=&quot]For example, if we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.


I can't find the solution can someone help me with this? this question is required for a forum registration and I suck at math XD

Thank you.[/FONT]

 

[ksdhart2]

One way to tackle it is just to "brute force" it. There's not too many possibilities. Counting them all would take a while but it shouldn't be too onerous a task. The general form of the numbers you want to find is 5x + 11y < 551. Let's step through a couple and see what we find.

x = 1, y = 0; 5(1) + 11(0) = 5
x = 2; y = 0; 5(2) + 11(0) = 10
x = 0; y = 1; 5(0) + 11(1) = 11
x = 3; y = 0; 5(3) + 11(0) = 15
x = 1; y = 1; 5(1) + 11(1) = 16
x = 4; y = 0; 5(4) + 11(0) = 20
...

If you keep going, you will get them all, and then the summing is straightforward. But, along the way you might find some patterns and shortcuts to help you out. For instance, consider just the multiples of 5. You know the highest such number is going to be 550. So part of the sum will be: 5 + 10 + 15 +... + 550. All of those numbers are divisible by 5, so you can factor that out, leaving: 5(1 + 2 + 3 + ... + 110). Where does that lead you? You might find this page helpful: ""The sum of the first n natural numbers"http://www.9math.com/book/sum-first-n-natural-numbers.

Please share with us any and all work you've done on this problem, even the parts you know for sure are wrong. The more specific you can be about exactly where you're getting bogged down, the better help we can provide. Thank you.

 

[JeffM]

Find the sum of the multiples of 11 or 5 under 551
For example, if we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.


I can't find the solution can someone help me with this? this question is required for a forum registration and I suck at math XD

Thank you.

The largest multiple of 5 under 551 is 550 = 110 * 5. So the multiples' sum is 5 * 1 + 5 * 2 * ... 5 * 110.

The largest multiple of 11 under 551 is also 550 = 50 * 11. So the multiples' sum is 11 * 1 + 11 * 2 + ... 11 * 50.

The number of multiples of 5 under 551 sum to \(\displaystyle 5 * \dfrac{110 * 111}{2} = 30525.\)

The number of multiples of 11 under 551 sum to \(\displaystyle 11 * \dfrac{50 * 51}{2} = 14025.\)

However, both the preceding sums include products of 5 and 11. Starting at 55 there are 10 of those under 551

Their sum is 55 * 1 + 55 * 2 + ... 55 * 10.

So we must adjust downwards by \(\displaystyle 55 * \dfrac{10 * 11}{2} = 3025.\)

The grand total is \(\displaystyle 30525 + 14025 - 3025 = 41525.\)

This is an example of counting principles and is not basic arithmetic.