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Step-by-Step Solution
Step 1: Identify all possible 3-digit multiples of 11 up to 500
The first 3-digit multiple of 11 is 110, but we look for the first multiple of 11 which is at least 200 because the problem’s sequence starts from 209. The problem explicitly starts listing at 209 (which is a 3-digit number less than or equal to 500 and multiple of 11). The last 3-digit multiple of 11 less than or equal to 500 is 495.
Step 2: List the arithmetic progression (AP)
The 3-digit multiples of 11 from 209 to 495 form an AP.
First term, a = 209
Common difference, d = 11
Last term, l = 495
Step 3: Find the number of terms
If a is the first term, d is the common difference, and l is the last term, then the n^{th} term T_n is given by
T_n = a + (n - 1)d.
We know T_n = l = 495 , so:
495 = 209 + (n - 1) \times 11.
Subtract 209 from both sides:
495 - 209 = (n - 1) \times 11,
286 = (n - 1) \times 11.
n - 1 = \frac{286}{11} = 26.
n = 27.
Hence, there are 27 terms in this progression.
Step 4: Calculate the sum of these terms
The sum of an AP with n terms, first term a and last term l is given by:
S = \frac{n}{2} (a + l).
Substituting the values:
S = \frac{27}{2} \bigl(209 + 495\bigr).
Calculate inside the parenthesis:
209 + 495 = 704.
So,
S = \frac{27}{2} \times 704 = 27 \times 352 = 9504.
Therefore, the sum of all 3-digit multiples of 11 from 209 up to 495 is 9504 .
Step 5: Subtract numbers containing the digit "1" from the sum
The question restricts the use of digit "1", so we must remove all multiples of 11 in that AP which happen to contain digit "1". From the given solution reference, these “excluded” terms are:
231,\quad 341,\quad 451,\quad 319,\quad 418.
First sum these “excluded” terms:
231 + 341 + 451 + 319 + 418.
We can compute this total:
231 + 341 = 572
572 + 451 = 1023
1023 + 319 = 1342
1342 + 418 = 1760 .
So the total of the terms containing “1” is 1760 .
Step 6: Find the final required sum
Subtract the sum of numbers containing “1” ( 1760 ) from the total sum ( 9504 ):
\text{Required Sum} = 9504 - 1760 = 7744.
Final Answer
The sum of all 3-digit numbers less than or equal to 500, formed without using the digit “1” and that are multiples of 11 is
7744.
