The number of three digit numbers which are multiples of 9 are

## Concept & Formula This problem can be solved by modeling the multiples of 9 as an Arithmetic Progression (A.P.). The formula for finding the $n$-th term of an A.P. is: $$T_n = a + (n - 1)d$$ Where $T_n$ is the last term, $a$ is the first term, and $d$ is the common difference. ## Step-by-Step Solution * **Given:** We need to find all 3-digit multiples of $9$. * The smallest 3-digit number is $100$. Dividing $100$ by $9$ gives a remainder of $1$. The first multiple is $100 + (9 - 1) = 108$. * The largest 3-digit number is $999$. Since $9 + 9 + 9 = 27$ (divisible by 9), $999$ itself is the last multiple. * **Calculation:** Set up the variables for the Arithmetic Progression. First term ($a$) = $108$ Last term ($T_n$) = $999$ Common difference ($d$) = $9$ * Apply the A.P. formula: $$999 = 108 + (n - 1)9$$ $$891 = (n - 1)9$$ $$n - 1 = \frac{891}{9}$$ $$n - 1 = 99$$ $$n = 100$$ ## Exam Strategy & Shortcut A much faster way to find the total count of multiples in a continuous range starting from 1 is simple division. * Total multiples of 9 from 1 to 999: $$\frac{999}{9} = 111$$ * Total multiples of 9 from 1 to 99 (the 1 and 2 digit numbers): $$\frac{99}{9} = 11$$ * To find strictly the 3-digit multiples, subtract the two values: $$111 - 11 = 100$$ This requires almost zero calculation time. ## Common Pitfall When using the A.P. formula, a common error is forgetting to add $1$ at the final step, leading students to select $99$ as the answer. When finding the number of terms between boundaries, it is an inclusive count. ## Final Answer Therefore, the correct answer is 100.