How to Prepare for Quantitatve Aptitude, Chapter: Number System, Topic: HCF & LCM

Question

Which of the following represents the largest 4 digit number which can be added to 7249 in order to make the derived number divisible by each of 12, 14, 21, 33, and 54.

(a) 9123

(b) 9383

(c) 8727

(d) None of these

Comments

$9383$

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I saw the solution. It has following equation

7249 +n = 8316*2

8316 is the LCM of 12,14,21,33, and 54.

I'm not able to understand the reason for multiplying 8316 with 2.

Answer 1

LCM of $12,14,21,33,54$

Or, LCM of $2 \times 2 \times 3,$    $2 \times 7,$    $3 \times 7,$    $3 \times 11,$    $2 \times 3 \times 3 \times 3$.

Or, LCM of $12,14,21,33,54$  is  $2^2 \times 3^3 \times 7 \times 11$ i.e. $8316$

Now, the derived number should be a multiple of $8316$ in order to get derived by $12.14.21.33.54.$

∴ $8316 \times 1 = 8316$    &  $8316-7249 = 1067$

   $8316 \times 2 = 16632$  &  $16632-7249 = 9383$

   $8316 \times 3 = 24948$  &  $24948-7249 = 17699$

   $8316 \times 4 = 33264$  &  $33264-7249 = 26015$

   $8316 \times 5 = 41580$  &  $41580-7249 = 34331$

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So, $1067$ is the smallest $4-digit$ number to be added with $7249$ in order to divide the number by $12,14,21,33,54$

$9383$ is the largest $4-digit$ number to be added with $7249$ in order to divide the number by $12,14,21,33,54.$

Now, we can observe that after $9383$, the numbers are not $4-digit$, they are $5-digit$ numbers.

∴ The answer will be $\color{purple}{9383}- \color{maroon}{\text{option B)}}$

Comments

thanks :)