The least number which when increased by $5$ is divisible by each one of $24$, $32$, $36$ and $54$ is

### Concept & Logic Let the required number be $x$. The question states that if we increase (add) $5$ to this number, it becomes perfectly divisible by $24$, $32$, $36$, and $54$. This means $(x + 5)$ is exactly the Least Common Multiple (LCM) of these divisors. Therefore, the core logic is: $$x = \text{LCM}(a, b, c, \dots) - \text{Increased Value}$$ ### Step-by-Step Solution * **Given:** Divisors are $24$, $32$, $36$, and $54$. Value to increase is $5$. * **Find the LCM:** Prime factorization: $24 = 2^3 \times 3^1$ $32 = 2^5$ $36 = 2^2 \times 3^2$ $54 = 2^1 \times 3^3$ Take the highest powers of all prime factors: $\text{Highest power of } 2 = 2^5 = 32$ $\text{Highest power of } 3 = 3^3 = 27$ $$LCM = 32 \times 27$$ To multiply quickly: $32 \times (30 - 3) = 960 - 96 = 864$. So, $\text{LCM} = 864$. * **Find the required number:** Since adding $5$ gets us to the LCM: $$\text{Required Number} = 864 - 5 = 859$$ ### Exam Strategy & Shortcut **Divisibility Rule Elimination:** One of the divisors is $54$, which is a multiple of $9$. Therefore, $(Option + 5)$ must be divisible by $9$. Let's test the options using the sum of digits rule for $9$: (a) $427 + 5 = 432 \rightarrow 4+3+2 = 9$ (Divisible) (b) $859 + 5 = 864 \rightarrow 8+6+4 = 18$ (Divisible) (c) $869 + 5 = 874 \rightarrow 8+7+4 = 19$ (Not divisible) (d) $4320 + 5 = 4325 \rightarrow 4+3+2+5 = 14$ (Not divisible) We are left with $427$ and $859$. Now test divisibility by $32$ (which means $(Option+5)$ must be divisible by $32$). $432 \div 32 = 13.5$ (Not divisible). Therefore, $859$ is the only valid answer. ### Common Pitfall Similar to the previous question type, students often do the reverse operation—they add $5$ to the LCM instead of subtracting it. Because the final number *needs* $5$ added to it to reach the LCM, the original number must be strictly *smaller* than the LCM. ### Final Answer **Therefore, the correct answer is 859.**