Sequences & Series
Sequences and Series
nta_pyq_2025_jan
Grade 11
Question:
The number of $3$-digit numbers, that are divisible by $2$ and $3$, but not divisible by $4$ and $9$, is \rule{2cm}{0.4pt}.
Step-by-Step Solution
Key Concept: Divisible by $2$ and $3$ $\Leftrightarrow$ divisible by $\operatorname{lcm}(2,3)=6$. Divisible by $4$ and $9$ $\Leftrightarrow$ divisible by $\operatorname{lcm}(4,9)=36$. Count multiples of $6$ minus multiples of $36$ in $[100,999]$.
Number of three-digit multiples of $6$: from $102$ to $996$.
$$N_{6}=\frac{996-102}{6}+1=149+1=150.$$
Number of three-digit multiples of $36$: from $108$ to $972$.
$$N_{36}=\frac{972-108}{36}+1=24+1=25.$$
Answer: $150-25=125.$
Correct Answer: 125