QuestionFind the GCD of 42 and 105. The GCD of 42 and 105 is (Type a whole number.)
Studdy Solution
STEP 1
Assumptions1. We are given two numbers42 and105. . We need to find the greatest common divisor (GCD) of these numbers.
STEP 2
The GCD of two numbers can be found using the Euclidean algorithm. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number.
For two numbers, a and b, where a > b, we can write it aswhere q is the quotient and r is the remainder. The GCD of a and b is the same as the GCD of b and r.
STEP 3
Let's start by dividing the larger number (105) by the smaller number (42) and find the remainder.
STEP 4
Calculate the quotient and the remainder.
So, q =2 and r =21.
STEP 5
Now, replace the larger number with the smaller number and the smaller number with the remainder and repeat the process.
STEP 6
Calculate the quotient and the remainder.
So, q =2 and r =0.
STEP 7
The Euclidean algorithm states that when the remainder is0, the divisor at this stage will be the greatest common divisor of the original two numbers. So, in this case, the GCD of42 and105 is21.
The greatest common divisor of42 and105 is21.
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