Finding the Sum of Natural Numbers Less Than 500 Divisible by 7 but Not 11

This article explains the methodology to find the sum of all natural numbers less than 500 that are divisible by 7 but not by 11. The process involves systematic steps to identify and exclude certain numbers based on specific divisibility criteria.

Step 1: Identifying Numbers Divisible by 7

To begin, we identify the largest natural number less than 500 that is divisible by 7. This can be determined by performing integer division on 499 by 7:

n leftlfloor frac{499}{7} rightrfloor 71

The largest multiple of 7 less than 500 is:

7 times 71 497

The natural numbers less than 500 that are divisible by 7 can be expressed as:

7, 14, 21, ldots, 497

This sequence forms an arithmetic series where:

First term (a) 7 Common difference (d) 7 Last term (l) 497

To find the number of terms (n) in this series, we use the formula for the n-th term of an arithmetic series:

l a (n-1)d

Rearranging the equation to solve for n:

497 - 7 (n-1) times 7 490 (n-1) times 7 n-1 frac{490}{7} 70 implies n 71

Step 2: Calculating the Sum of the Series

The sum (S) of the first n terms of an arithmetic series is given by:

S_n frac{n}{2} times (a l)

Plugging in the values:

S_{71} frac{71}{2} times (7 497) frac{71}{2} times 504 71 times 252 17892

Step 3: Excluding Numbers Divisible by 11

Next, we find the numbers that are divisible by both 7 and 11, which means looking for numbers divisible by 77 (since 77 7 times 11):

The largest natural number less than 500 that is divisible by 77 is:

m leftlfloor frac{499}{77} rightrfloor 6

The largest multiple of 77 less than 500 is:

77 times 6 462

The natural numbers less than 500 that are divisible by 77 can be expressed as:

77, 154, 231, 308, 385, 462

This sequence also forms an arithmetic series where:

First term (a) 77 Common difference (d) 77 Last term (l) 462

To find the number of terms (m) in this series:

462 - 77 (m-1) times 77 385 (m-1) times 77 m-1 frac{385}{77} 5 implies m 6

Calculate the sum of this series:

S_m frac{m}{2} times (a l) frac{6}{2} times (77 462) 3 times 539 1617

Final Calculation

The final step is to subtract the sum of the multiples of 77 from the sum of the multiples of 7:

text{Sum of multiples of 7 but not 11} S_{71} - S_6 17892 - 1617 16275

Hence, the sum of all natural numbers less than 500 that are divisible by 7 but not by 11 is:

boxed{16275}

Conclusion

By following this systematic approach, we have identified and summed up all natural numbers less than 500 that meet the specified criteria, resulting in the final sum of 16275. This method can be applied to similar problems involving divisibility and summation.