Question

12. Find the highest positive integer which leaves the same remainder on dividing the
numbers 195, 303, 327

Answer 1

Answer:

12

Step-by-step explanation:

Let the same remainder which is obtained on dividing 195, 303, 327 be r

So, The no. that are exactly divisible by highest positive integer are :

• ( 195 - r )
• ( 303 - r )
• ( 327 - r )

Required highest positive integer will ne the HCF of ( 195 - r ) , ( 303 - r ) and ( 327 - r)

We know that

If a few no.s are divisible by a no. then if we found differences between them or sums the resultant no.s will also be divisible by the same no.

So. HCF of { ( 303 - r) - ( 195 - r) } and { ( 327 - r) - ( 303 - r) } is the required highest positive integer

• { ( 303 - r) - ( 195 - r) } = 303 - 195 = 108

• { ( 327 - r) - ( 303 - r) } = 24

Now we will calculate the HCF of 108 and 24 using Euclid algorithm a = bq + r, where 0 < or = r < b

108 = 24 × 4 + 12

24 = 12 × 2 + 0

Since reminder is 0 divisor at this stage becomes HCF

=> HCF of 108 and 24 = 12

=> HCF of ( 195 - r) , (303 - r) and (327 - r) = 12

Therefore the required highest positive integer is 12.