The sum of all integers from 1 to 520 that are multiples of 4 or 5?ans.20842118752This Questi

1. Break Down the Problem

To find the sum of all integers from 1 to 520 that are multiples of 4 or 5, we can break the problem into the following parts:

1. Find the sum of multiples of 4 from 1 to 520.
2. Find the sum of multiples of 5 from 1 to 520.
3. Subtract the sum of multiples of both 4 and 5 (i.e., multiples of 20) to avoid double counting.

2. Relevant Concepts

We will use the formula for the sum of an arithmetic series: $S_n = \frac{n}{2} (a + l)$ where:

• $S_n$ = sum of the series,
• $n$ = number of terms,
• $a$ = first term,
• $l$ = last term.

3. Analysis and Detail

1. Sum of Multiples of 4:

   • The multiples of 4 up to 520 are $4, 8, 12, \ldots, 520$.
   • This is an arithmetic series where $a = 4$ and $l = 520$.
   • To find $n$: $n = \frac{520 - 4}{4} + 1 = 130$
   • Now, calculate the sum: $S_4 = \frac{130}{2} (4 + 520) = 65 \times 524 = 34160$

2. Sum of Multiples of 5:

   • The multiples of 5 up to 520 are $5, 10, 15, \ldots, 520$.
   • Here, $a = 5$ and $l = 520$.
   • To find $n$: $n = \frac{520 - 5}{5} + 1 = 104$
   • Now, calculate the sum: $S_5 = \frac{104}{2} (5 + 520) = 52 \times 525 = 27300$

3. Sum of Multiples of 20:

   • The multiples of 20 up to 520 are $20, 40, 60, \ldots, 520$.
   • Here, $a = 20$ and $l = 520$.
   • To find $n$: $n = \frac{520 - 20}{20} + 1 = 26$
   • Now, calculate the sum: $S_{20} = \frac{26}{2} (20 + 520) = 13 \times 540 = 7020$

4. Verify and Summarize

Now, we can combine the results:

• Total sum is: $\text{Total Sum} = S_4 + S_5 - S_{20} = 34160 + 27300 - 7020 = 54440$

Final Answer

The sum of all integers from 1 to 520 that are multiples of 4 or 5 is 54440.