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Find the greatest common divisor of 34529 and 32923.

The greatest common divisor is the largest number that divides exactly into both of our numbers. This is also often referred to as the highest common factor. One way of working out the greatest common divisor, or highest common factor of two large numbers, is using prime factorization. Prime numbers have exactly two factors, the number one and the number itself. The prime numbers that are less than 50 are as follows. Two, three, five, seven, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. All of these numbers are only divisible by the number one and the number itself.

Let’s consider our first number, 34529. As this is an odd number, it will not be divisible by two. The digits of 34529 sum to 23. As this is not a multiple of three, then 34529 will not be divisible by three. Any whole number is divisible by three if the sum of its digits are also divisible by three. A number is divisible by five if it ends in five or zero. As 34529 does not, it will not be divisible by five. 34529 is not divisible by seven.

We could check this using the short division bus stop method if we did not have a calculator. As three is not divisible by seven, we carry the three to the thousands column. 34 divided by seven is equal to four remainder six. 65 divided by seven is equal to nine remainder two. 22 divided by seven is equal to three remainder one. Finally, 19 divided by seven is equal to two remainder five. As there is a remainder, we have proved that 34529 is not divisible by seven.

Our next prime number is 11. We need to check whether 34529 is divisible by 11. Three is not divisible by 11. 34 divided by 11 is equal to three remainder one. 15 divided by 11 is equal to one remainder four. 42 divided by 11 is equal to three remainder nine. Finally, 99 divided by 11 is equal to nine. This means that 11 is a factor of 34529. 11 multiplied by 3139 is equal to 34529. As 11 is a prime number, we circle it. We now need to repeat the process to find a prime factor of 3139. 3139 is not divisible by any of the prime numbers 13, 17, 19, 23, 29, 31, 37, or 41. It is, however, divisible by 43. 43 multiplied by 73 is equal to 3139. Both of these are prime numbers. Therefore, we need to circle both of them.

We can check this answer by multiplying 43 by 73. This can be done using the grid method. We split 43 into 40 and three. And we split 73 into 70 and three. 40 multiplied by 70 is equal to 2800. As four multiplied by seven is 28, and we add the two zeros. 40 multiplied by three is equal to 120. Three multiplied by 70 is equal to 210, as three multiplied by seven is 21. Finally, three multiplied by three is equal to nine. Adding the top two numbers gives us 2920. Adding the bottom two numbers gives us 219. These two numbers added together gives us 3139. Therefore, our calculation is correct. The number 34529 can be written as a product of its prime factors, as 11 multiplied by 43 multiplied by 73.

We now need to repeat this process for our second number, 32923. Once again, this number is not divisible by two, three, five, or seven. It is divisible by 11. 11 multiplied by 2993 is equal to 32923. The next prime factor of 2993 is 41. 41 multiplied by 73 is equal to 2993. As both of these are prime numbers, 32923 can be written as a product of its prime factors, as 11 multiplied by 41 multiplied by 73.

In order to work out the greatest common divisor, or highest common factor, we need to look at which prime factors are common. Both of these numbers have prime factors of 11 and 73. This means that the greatest common divisor is equal to 11 multiplied by 73. 10 multiplied by 73 is 730. One multiplied by 73 is 73. Adding these numbers gives us the answer to 11 multiplied by 73. This is equal to 803. Therefore, the greatest common divisor of 34529 and 32923 is 803.