The sum of all three-digit numbers less than or equal to 500, that are formed without usin…

Question

The sum of all three-digit numbers less than or equal to 500, that are formed without using the digit 1 and they all are multiples of 11, is:

Answer: 7744

Accepted Answer

3-digit multiples of 11, less than 500: $121,132,\ldots,495$, count $=\dfrac{495-121}{11}+1=35$, sum $=\dfrac{35}{2}(121+495)=10780$.

Now subtract those containing digit 1 (which must be excluded).

Multiples of 11 (< 500) with 1 in the hundreds place: $121$ to $198$, count $8$, sum $=\dfrac82(121+198)=1276$.

With 1 in the tens place (hundreds $\ne1$): $319,418$, sum $=737$.

With 1 in the units place (hundreds, tens $\ne1$): $231,341,451$, sum $=1023$.

$$\text{Required sum} = 10780 - 1276 - 737 - 1023 = 7744$$

Solutions · 1