Video: Solving Word Problems Involving the Highest Common Factor

To perform the never ending handkerchief magic trick, a magician wants to cut a yellow silk fabric of length 28 inches, a green one of length 36 inches, and a red one of length 46 inches into handkerchiefs of equal lengths. If he uses the total length of all the fabrics, determine the length of the longest handkerchief he can cut from each fabric, and then find how many handkerchiefs he can make in that case.

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Video Transcript

To perform the never-ending handkerchief magic trick, a magician wants to cut a yellow silk fabric of length 28 inches, a green one of length 36 inches, and a red one of length 46 inches into handkerchiefs of equal lengths. If he uses the total length of all the fabrics, determine the length of the longest handkerchief he can cut from each fabric and then find out how many handkerchiefs he can make in that case.

The magician has three coloured fabrics, yellow 28 inches long, green 36 inches long, and red 46 inches long. As he wants to cut them into pieces of equal length, we need to work out the highest common factor of 28, 36, and 46. This is the highest number that divides exactly into all three numbers. One way of finding all the factors of each of the numbers is using prime factorization. Prime numbers have exactly two factors, one and the number itself. The first 10 of them are two, three, five, seven, 11, 13, 17, 19, 23, and 29. Lets firstly, consider the yellow fabric of length of 28 inches.

As 28 is an even number, it is divisible by two. Two multiplied by 14 is equal to 28. 14 is also divisible by two, as two multiplied by seven is 14. Both of these are prime numbers. This means that 28, written as a product of its prime factors, is two multiplied by two multiplied by seven. We can use the same method for the green fabric. This was 36 inches long. 36 is equal to two multiplied by 18. 18 is equal to two multiplied by nine. And nine is equal to three multiplied by three. This means that 36 is equal to two multiplied by two multiplied by three multiplied by three.

Finally, let’s consider the red fabric of length 46 inches. Once again, this is divisible by two as 46 is an even number. A half of 46 is 23. Therefore, two multiplied by 23 is 46. 23 is also a prime number as it only has two factors. Therefore, 46 written as a product of its prime factors is two multiplied by 23. The only common prime factor of 28, 36, and 46 is two. This means that the highest common factor of 28, 36, and 46 is also two. In order for the magician to cut the handkerchiefs into equal lengths, they need to be of length two inches. For the yellow fabric, there will be 14 times two inches, for the green fabric 18 times two inches, and for the red fabric 23 multiplied by two inches.

14 plus 18 is equal to 32. Adding 23 to this gives us 55. In total, the magician will have 55 handkerchiefs of length two inches. 55 multiplied by two is equal to 110. We can check that 28 inches plus 36 inches plus 46 inches does indeed equal 110 inches. As this is indeed correct. We know that our answer of 55 handkerchiefs of length two inches is right.

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