The H.C.F. of $204$, $1190$ and $1445$ is

### Concept & Strategy For finding the HCF of three large numbers, the continuous division method can be lengthy. The most efficient approach is the **Difference Method**. The HCF of any set of numbers will always evenly divide the difference between any two of those numbers. By finding the difference between the two numbers closest to each other, we drastically reduce the pool of possible factors. ### Step-by-Step Solution * **Given:** The numbers are $204, 1190$, and $1445$. * **Calculation / Deduction:** 1. Identify the two numbers with the smallest gap between them. Here, they are $1190$ and $1445$. 2. Calculate their difference: $$1445 - 1190 = 255$$ 3. The HCF of the three numbers *must* be a factor of $255$. 4. Let's look at the given options: $17, 18, 19, 21$. 5. Since $255$ ends in $5$, it is an odd number. Therefore, it cannot have any even factors. This immediately eliminates Option (b) $18$. 6. Let's test if $255$ is divisible by Option (a) $17$: $$255 \div 17 = 15$$ Since $17 \times 15 = 255$, $17$ is a valid factor of the difference. 7. To be absolutely sure, verify if $17$ divides the smallest original number, $204$: $$204 \div 17 = 12$$ 8. Since $17$ divides both the difference of the large numbers and the standalone small number perfectly, it is our HCF. ### Exam Strategy & Shortcut **Unit Digit Viability Check:** Take the smallest number, $204$, and test the options using unit digits. We need an option that, when multiplied by some integer, ends in $4$. * (b) $18$: $8 \times 3 = 24$, $8 \times 8 = 64$. Possible, but $204/18$ fails. * (c) $19$: $9 \times 6 = 54$. $19 \times 10 = 190$, plus $19$ is $209$. Fails. * (d) $21$: $1 \times 4 = 4$. $21 \times 10 = 210$, so $21 \times 9 = 189$. Fails. * (a) $17$: $7 \times 2 = 14$. Let's try a multiplier ending in $2$, like $12$. $17 \times 10 = 170$. $17 \times 2 = 34$. $170 + 34 = 204$. Perfect match. ### Common Pitfall Attempting to apply prime factorization or the ladder division method to $1190$ and $1445$. These numbers have large, obscure prime factors (like 17) which are difficult to spot mentally, leading to a massive loss of time during an exam. Always use the difference method for large values. ### Final Answer **Therefore, the correct answer is $17$.**