Find the greatest common divisor of negative 2805 and 1287.
The greatest common divisor of the two numbers is the largest number that divides exactly into both of them. This is also sometimes known as the highest common factor. The fact that the first number is negative will not impact our answer. As all positive numbers are greater than negative numbers, our answer will have to be positive. We are therefore looking for the highest number that divides exactly into 2805 and 1287.
The easiest way of doing this is using prime factorization. Prime numbers have exactly two factors. The number one and the number itself. The prime numbers that are less than 20 are two, three, five, seven, 11, 13, 17, and 19. We need to split the numbers 2805 and 1287 into the product of their prime factors.
2805 is not divisible by two, as it is not an even number. We know that any number that ends in five or zero is divisible by five. Therefore, 2805 will be divisible by five. We could do this calculation using a short division, bus stop method. Two is not divisible by five, so we need to carry the two to the hundreds column. 28 divided by five is equal to five remainder three. 30 divided by five is equal to six. And, five divided by five is equal to one. This means that 2805 divided by five is 561. We can therefore say that five multiplied by 561 is equal to 2805. As five is a prime number, we circle it.
Next, we need to find a prime number that divides exactly into 561. Let’s consider the number three. Any number is divisible by three if the sum of its digits is also divisible by three. The digits of 561 sum to 12. And, 12 is divisible by three. Therefore, 561 will also be divisible by three. Once again, we can work this out using the bus stop method.
Five divided by three is equal to one remainder two. 26 divided by three is equal to eight remainder two. And finally, 21 divided by three is equal to seven. Therefore, 561 divided by three is 187. We can split 561 into three and 187. 187 is not divisible by two, three, five, or seven. However, it is divisible by 11, as 11 multiplied by 17 is 187. 10 multiplied by 17 is 170. So, 11 multiplied by 17 must be 187. As both of these numbers are prime, we need to circle them both.
2805 written as a product of its prime factors is three multiplied by five multiplied by 11 multiplied by 17. We now need to repeat this process for the number 1287. Once again, as it is an odd number, it will not be divisible by two. It is, however, divisible by three. Three multiplied by 429 is 1287. 429 is also divisible by three. Three multiplied by 143 is equal to 429. 143 is not divisible by two, three, five, or seven. It is divisible by 11. 11 multiplied by 13 is equal to 143. This means that 1287, written as a product of its prime factors, is three multiplied by three multiplied by 11 multiplied by 13.
We want to find the highest common factor or greatest common divisor. Therefore, we are looking for factors that are common to both numbers. Three and eleven are factors of both numbers. This means that the greatest common divisor is equal to three multiplied by 11. This is equal to 33. Therefore, the greatest common divisor of 2805 and 1287 is 33. This is also the case for negative 2805 and 1287. 33 multiplied by negative 85 is equal to negative 2805. And, 33 multiplied by 39 is equal to 1287.