Question

Find out the greatest number by which 1709 and 2007 can be divided leaving the remainders 7 and 9
respectively.​

Answer 1

Answer:

We need to find the HCF in this question .....

Subtracting the remainders 7 & 9 from 1709 and 2007 respectively

we get , 1702 & 1998

FINDING THEIR HCF

1702 = 2 × 37 × 23

1998 = 2 × 3 × 3 × 3 × 37

Therefore HCF 1702 and 1998 = 2 × 37

= 74

And = 74

Answer 2

Concept: The biggest positive number d such that d is a divisor of both a and b, i.e., there are integers e and f such that a = de and b = df, and d is the largest such integer, is the greatest common divisor (GCD) of two nonzero integers a and b. The standard representation for the GCD of a and b is gcd (a, b).

Given: Two numbers are given 1709 and 2007.

Find: Two find the greatest number by which 1709 and 2007 can be divided leaving the remainder 7 and 9 respectively.

Solution:

Let x be that number then

1709 = x×t+7 for an integer t

2007 = x×s+9 for an integer s

So, x is the GCD(1709-7, 2007-9) = GCD(1702, 1998)

We will now find the GCD of 1702 and 1998.

We have 1702 =  2 × 37 × 23

1998 = 2 × 37× 3 × 3 × 3

So, the greatest common divisor is 37 × 2 = 74