The smallest number which when diminished by $7$, is divisible by $12$, $16$, $18$, $21$ and $28$ is

### Concept & Logic Let the required number be $x$. The question states that if we diminish (subtract) $7$ from this number, the result becomes perfectly divisible by $12$, $16$, $18$, $21$, and $28$. This means $(x - 7)$ must be exactly the Least Common Multiple (LCM) of these given numbers. Therefore, to find $x$, the formula is: $$x = \text{LCM}(a, b, c, \dots) + \text{Remainder/Diminished Value}$$ ### Step-by-Step Solution * **Given:** Divisors are $12$, $16$, $18$, $21$, and $28$. The value to diminish is $7$. * **Find the LCM:** Let's break them down into prime factors: $12 = 2^2 \times 3^1$ $16 = 2^4$ $18 = 2^1 \times 3^2$ $21 = 3^1 \times 7^1$ $28 = 2^2 \times 7^1$ The LCM requires the highest power of every prime factor present: $\text{Highest power of } 2 = 2^4 = 16$ $\text{Highest power of } 3 = 3^2 = 9$ $\text{Highest power of } 7 = 7^1 = 7$ $$LCM = 16 \times 9 \times 7 = 144 \times 7 = 1008$$ * **Find the required number:** The LCM ($1008$) is the number that is exactly divisible by all of them. But our target number, when reduced by $7$, becomes $1008$. $$\text{Required Number} = 1008 + 7 = 1015$$ ### Exam Strategy & Shortcut **Divisibility Rule of 9:** The divisors include $18$. For a number to be divisible by $18$, it must be divisible by $9$. This means $(Option - 7)$ must be divisible by $9$. Let's apply the sum of digits rule for $9$: (a) $1008 - 7 = 1001 \rightarrow$ Sum = $2$ (Not divisible) (b) $1015 - 7 = 1008 \rightarrow$ Sum = $9$ (Divisible by $9$) (c) $1022 - 7 = 1015 \rightarrow$ Sum = $7$ (Not divisible) (d) $1032 - 7 = 1025 \rightarrow$ Sum = $8$ (Not divisible) You found the answer in 10 seconds without calculating the LCM! ### Common Pitfall The most common mistake is subtracting $7$ from the LCM instead of adding it. Always translate the English condition to a basic algebraic equation: "Number diminished by $7$ equals LCM" becomes $N - 7 = \text{LCM}$, which clearly shows $N = \text{LCM} + 7$. ### Final Answer **Therefore, the correct answer is 1015.**