How do you explain GCF in mathematics?
GCF or greatest common factor of two or more numbers is defined as largest possible number or integer which is the factor of all given number or in other words we can say that largest possible common number which completely divides the given numbers. GCF of two numbers can be represented as GCF (84, 140).
Properties of GCF
- The GCF of two or more given numbers cannot be greater than any of the given number. Eg- GCF of 84 and 140 is 28, where 28 is less than both 84 and 140.
- GCF of two consecutive numbers is always 1.
- The product of GCF and LCM of two given numbers is equal to the product of two numbers.
- The GCF of two given numbers where one of them is a prime number is either 1 or the number itself.
How can we define factors?
In mathematics, a factor is a number which divides into another number exactly, without leaving any remainder. A factor of a number can be positive of negative.
Properties of Factors
- Each number is a factor of itself. Eg. 84 and 140 are factors of themselves respectively.
- 1 is a factor of every number. Eg. 1 is a factor of 84 and also of 140.
- Every number is a factor of zero (0), since 84 x 0 = 0 and 140 x 0 = 0.
- Every number other than 1 has at least two factors, namely the number itself and 1.
- Every factor of a number is an exact divisor of that number, example 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 are exact divisors of 84 and 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140 are exact divisors of 140.
- Factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Each factor divides 84 without leaving a remainder.
Simlarly, factors of 140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140. Each factor divides 140 without leaving a remainder. - Every factor of a number is less than or equal to the number, eg. 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 are all less than or equal to 84 and 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140 are all less than or equal to 140.
Steps to find Factors of 84 and 140
- Step 1. Find all the numbers that would divide 84 and 140 without leaving any remainder. Starting with the number 1 upto 42 (half of 84) and 1 upto 70 (half of 140). The number 1 and the number itself are always factors of the given number.
84 ÷ 1 : Remainder = 0
140 ÷ 1 : Remainder = 0
84 ÷ 2 : Remainder = 0
140 ÷ 2 : Remainder = 0
84 ÷ 3 : Remainder = 0
140 ÷ 4 : Remainder = 0
84 ÷ 4 : Remainder = 0
140 ÷ 5 : Remainder = 0
84 ÷ 6 : Remainder = 0
140 ÷ 7 : Remainder = 0
84 ÷ 7 : Remainder = 0
140 ÷ 10 : Remainder = 0
84 ÷ 12 : Remainder = 0
140 ÷ 14 : Remainder = 0
84 ÷ 14 : Remainder = 0
140 ÷ 20 : Remainder = 0
84 ÷ 21 : Remainder = 0
140 ÷ 28 : Remainder = 0
84 ÷ 28 : Remainder = 0
140 ÷ 35 : Remainder = 0
84 ÷ 42 : Remainder = 0
140 ÷ 70 : Remainder = 0
84 ÷ 84 : Remainder = 0
140 ÷ 140 : Remainder = 0
Hence, Factors of
84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84
And, Factors of
140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140
Examples of GCF
Sammy baked 84 chocolate cookies and 140 fruit and nut cookies to package in plastic containers for her friends at college. She wants to divide the cookies into identical boxes so that each box has the same number of each kind of cookies. She wishes that each box should have greatest number of cookies possible, how many plastic boxes does she need?Since Sammy wants to pack greatest number of cookies possible. So for calculating total number of boxes required we need to calculate the GCF of 84 and 140.
GCF of 84 and 140 is 28.
A class has 84 boys and 140 girls. A choir teacher wants to form a choir team from this class such that the students are standing in equal rows also girls or boys will be in each row. Teacher wants to know the greatest number of students that could be in each row, can you help him?To find the greatest number of students that could be in each row, we need to find the GCF of 84 and 140. Hence, GCF of 84 and 140 is 28.
What is the difference between GCF and LCM?Major and simple difference betwen GCF and LCM is that GCF gives you the greatest common factor while LCM finds out the least common factor possible for the given numbers.
Ram has 84 cans of Pepsi and 140 cans of Coca Cola. He wants to create identical refreshment tables that will be organized in his house warming party. He also doesn't want to have any can left over. What is the greatest number of tables that Ram can arrange?To find the greatest number of tables that Ram can stock we need to find the GCF of 84 and 140. Hence GCF of 84 and 140 is 28. So the number of tables that can be arranged is 28.
Ariel is making ready to eat meals to share with friends. She has 84 bottles of water and 140 cans of food, which she would like to distribute equally, with no left overs. What is the greatest number of boxes Ariel can make?The greatest number of boxes Ariel can make would be equal to GCF of 84 and 140. So the GCF of 84 and 140 is 28.
Mary has 84 blue buttons and 140 white buttons. She wants to place them in identical groups without any buttons left, in the greatest way possible. Can you help Mary arranging them in groups?Greatest possible way in which Mary can arrange them in groups would be GCF of 84 and 140. Hence, the GCF of 84 and 140 or the greatest arrangement is 28.
Kamal is making identical balloon arrangements for a party. He has 84 maroon balloons, and 140 orange balloons. He wants each arrangement tohave the same number of each color. What is the greatest number of arrangements that he can make if every balloon is used?The greatest number of arrangements that he can make if every balloon is used would be equal to GCF of 84 and 140. So the GCF of 84 and 140 is 28.
Kunal is making baskets full of nuts and dried fruits. He has 84 bags of nuts and 140 bags of dried fruits. He wants each basket to be identical, containing the same combination of bags of nuts and bags of driesn fruits, with no left overs. What is the greatest number of baskets that Kunal can make?the greatest number of baskets that Kunal can make would be equal to GCF of 84 and 140. So the GCF of 84 and 140 is 28.
To energize public transportation, Abir needs to give a few companions envelopes with transport tickets, and metro tickets in them. On the off chance that he has 84 bus tickets and 140 metro tickets to be parted similarly among the envelopes, and he need no tickets left. What is the greatest number of envelopes Abir can make?To make the greatest number of envelopes Abir needs to find out the GCF of 84 and 140. Hence, GCF of 84 and 140 is 28.