Question : If 847 × 385 × 675 × 3025 = 3a × 5b × 7c × 11d, then the value of ab – cd is:
Option 1: 4
Option 2: 5
Option 3: 1
Option 4: 7
Correct Answer: 5
Solution :
$847 × 385 × 675 × 3025 = 3^\text{a} × 5^\text{b} × 7^\text{c} × 11^\text{d}$
We can write the above numbers as
$847 = 11 × 11 × 7 = 11^2 × 7^1$
$385 = 5^1 × 7^1 × 11^1$
$675 = 3 × 3 × 3 × 5 × 5 = 3^3 × 5^2$
$3025 = 5 × 5 × 11 × 11 = 5^2 × 11^2$
Now, we have
$847 × 385 × 675 × 3025 = 3^\text{a} × 5^\text{b} × 7^\text{c} × 11^\text{d}$
$⇒11^2 × 7^1 × 5^1 × 7^1 × 11^1 × 3^3 × 5^2 × 5^2 × 11^2 = 3^\text{a} × 5^\text{b} × 7^\text{c} × 11^\text{d}$
$⇒3^3 × 5^5 × 7^2 × 11^5= 3^\text{a} × 5^\text{b} × 7^\text{c} × 11^\text{d}$
On comparing we get
$\text{a} = 3, \text{b} = 5, \text{c} = 2$ and $\text{d} = 5$
Now, $\text{ab} - \text{cd} = (3 × 5 - 2 × 5) = (15 - 10) = 5$
Hence, the correct answer is 5.
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