Question

The largest number that can divide 147,183 and 271 leaving remainders 11,13 and 16, respectively, is:

• A. 14
• B. 17
• C. 15
• D. 16

Answer 1

Answer:

B. 17

Let the required greatest number be x.

If a number leaves remainders 11, 13 and 16 when dividing 147, 183 and 271 respectively, then:

$147 - 11,\quad 183 - 13,\quad 271 - 16$

must be divisible by x.

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Step 1: Subtract the remainders

$147 - 11 = 136$
$183 - 13 = 170$
$271 - 16 = 255$

So, x must divide 136, 170 and 255.

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Step 2: Find HCF (GCD) of 136, 170 and 255

First find GCD of 136 and 170:

$170 - 136 = 34$

Now find GCD of 136 and 34:

$136 ÷ 34 = 4 \text{ (exact)}$

So,
$\text{GCD}(136,170) = 34$

Now find GCD of 34 and 255:

$255 ÷ 34 = 7 \text{ remainder } 17$

Now GCD of 34 and 17:

$34 ÷ 17 = 2 \text{ (exact)}$

So,

$\text{HCF} = 17$

Final Answer:

$\boxed{17}$