QuestionFind the greatest common divisor (gcd) of 16 and 50. What is the gcd? (Type a whole number.)

Studdy Solution

STEP 1

Assumptions1. We are asked to find the greatest common divisor (GCD) of two numbers16 and50. . The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

STEP 2

We will use the Euclidean algorithm to find the GCD. The Euclidean algorithm is a way to find the greatest common divisor of two numbers by dividing the larger number by the smaller number, then dividing the remainder by the smaller number, and so on, until the remainder is zero. The GCD is the last non-zero remainder.
The Euclidean algorithm can be expressed as follows $gcd(a, b) = gcd(b, a \mod b)$

STEP 3

Let's start by applying the Euclidean algorithm to our numbers. We'll take50 as 'a' and16 as 'b'.
$gcd(50,16) = gcd(16,50 \mod16)$

STEP 4

Calculate the remainder when50 is divided by16.
$50 \mod16 =2$

STEP 5

Substitute the value of the remainder into the Euclidean algorithm.
$gcd(50,16) = gcd(16,2)$

STEP 6

Continue the process by replacing 'a' with 'b' and 'b' with 'a mod b'.
$gcd(16,2) = gcd(2,16 \mod2)$

STEP 7

Calculate the remainder when16 is divided by2.
$16 \mod2 =0$

STEP 8

Substitute the value of the remainder into the Euclidean algorithm.
$gcd(16,2) = gcd(2,0)$

STEP 9

The Euclidean algorithm stops when 'b' is. The GCD is the last non-zero remainder, which in this case is2.
The greatest common divisor of16 and50 is2.

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QuestionFind the greatest common divisor (gcd) of 16 and 50. What is the gcd? (Type a whole number.)

Studdy Solution

STEP 1

Assumptions1. We are asked to find the greatest common divisor (GCD) of two numbers16 and50. . The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

STEP 2

STEP 3

Let's start by applying the Euclidean algorithm to our numbers. We'll take50 as 'a' and16 as 'b'.
$gcd(50,16) = gcd(16,50 \mod16)$

STEP 4

Calculate the remainder when50 is divided by16.
$50 \mod16 =2$

STEP 5

Substitute the value of the remainder into the Euclidean algorithm.
$gcd(50,16) = gcd(16,2)$

STEP 6

Continue the process by replacing 'a' with 'b' and 'b' with 'a mod b'.
$gcd(16,2) = gcd(2,16 \mod2)$

STEP 7

Calculate the remainder when16 is divided by2.
$16 \mod2 =0$

STEP 8

Substitute the value of the remainder into the Euclidean algorithm.
$gcd(16,2) = gcd(2,0)$

STEP 9

The Euclidean algorithm stops when 'b' is. The GCD is the last non-zero remainder, which in this case is2.
The greatest common divisor of16 and50 is2.

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QuestionFind the greatest common divisor (gcd) of 16 and 50. What is the gcd? (Type a whole number.)

Studdy Solution

STEP 1

Assumptions1. We are asked to find the greatest common divisor (GCD) of two numbers16 and50. . The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

STEP 2

STEP 3

Let's start by applying the Euclidean algorithm to our numbers. We'll take50 as 'a' and16 as 'b'.gcd(50,16)=gcd(16,50mod 16)gcd(50,16) = gcd(16,50 \mod16)gcd(50,16)=gcd(16,50mod16)

STEP 4

Calculate the remainder when50 is divided by16.50mod 16=250 \mod16 =250mod16=2

STEP 5

Substitute the value of the remainder into the Euclidean algorithm.gcd(50,16)=gcd(16,2)gcd(50,16) = gcd(16,2)gcd(50,16)=gcd(16,2)

STEP 6

Continue the process by replacing 'a' with 'b' and 'b' with 'a mod b'.gcd(16,2)=gcd(2,16mod 2)gcd(16,2) = gcd(2,16 \mod2)gcd(16,2)=gcd(2,16mod2)

STEP 7

Calculate the remainder when16 is divided by2.16mod 2=016 \mod2 =016mod2=0

STEP 8

Substitute the value of the remainder into the Euclidean algorithm.gcd(16,2)=gcd(2,0)gcd(16,2) = gcd(2,0)gcd(16,2)=gcd(2,0)

STEP 9

The Euclidean algorithm stops when 'b' is. The GCD is the last non-zero remainder, which in this case is2.The greatest common divisor of16 and50 is2.

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Let's start by applying the Euclidean algorithm to our numbers. We'll take50 as 'a' and16 as 'b'.
$gcd(50,16) = gcd(16,50 \mod16)$

Calculate the remainder when50 is divided by16.
$50 \mod16 =2$

Substitute the value of the remainder into the Euclidean algorithm.
$gcd(50,16) = gcd(16,2)$

Continue the process by replacing 'a' with 'b' and 'b' with 'a mod b'.
$gcd(16,2) = gcd(2,16 \mod2)$

Calculate the remainder when16 is divided by2.
$16 \mod2 =0$

Substitute the value of the remainder into the Euclidean algorithm.
$gcd(16,2) = gcd(2,0)$

The Euclidean algorithm stops when 'b' is. The GCD is the last non-zero remainder, which in this case is2.
The greatest common divisor of16 and50 is2.