In 26 and 27, use the extended Euclidean algorithm to find the greatest common divisor of the given numbers and express it as a linear combination of the two numbers.

4158 and 1568.

To calculate:

The greatest common divisor of 4158 and 1568 and express it in a linear combination of two numbers.

Given information:

The given numbers are 4158 and 1568.

Formula used:

Division algorithm:

Let a be an integer and d be a positive integer. Then, there are unique integers q and r with 0≤r<d such that a=dq+r, here, q is the quotient and r is the remainder.

So, q=a div d and r=amodd.

The greatest common divisor of two numbers a and b is the integer d which satisfies the following properties:

d|a and d|b

For all integers c, when c|a and c|b, then c≤d.

Calculation:

To find the greatest common divisor, use the Euclidean algorithm.

Step 1: 4158=1568×2+1022⇒1022=4158−1568×2

Step 2: 1568=1022×1+546⇒546=1568−1022×1

Step 3: 1022=546×1+476⇒476=1022−546×1

Step 4: 546=476×1+70⇒70=546−476×1

Step 5: 476=70×6+56⇒56=476−70×6

Step 6: 70=56×1+14⇒14=70−56×1

Step 7: 56=14×4+0

So, 14 is the remainder.

gcd(4158,1568)=14

To write the gcd in a linear combination, use the step 6