Definition of GCF
Greatest common factor commonly known as GCF of the two numbers is the highest possible number which completely divides given numbers, i.e. without leaving any remainder. It is represented as GCF (120, 405).
Properties of GCF
- The GCF of two or more given numbers cannot be greater than any of the given number. Eg- GCF of 120 and 405 is 15, where 15 is less than both 120 and 405.
- 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.
What are factors?
In mathematics, a factor is that number which divides into another number exactly, without leaving a remainder. A factor of a number can be positive or negative.
Properties of Factors
- Each number is a factor of itself. Eg. 120 and 405 are factors of themselves respectively.
- 1 is a factor of every number. Eg. 1 is a factor of 120 and also of 405.
- Every number is a factor of zero (0), since 120 x 0 = 0 and 405 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, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 are exact divisors of 120 and 1, 3, 5, 9, 15, 27, 45, 81, 135, 405 are exact divisors of 405.
- Factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Each factor divides 120 without leaving a remainder.
Simlarly, factors of 405 are 1, 3, 5, 9, 15, 27, 45, 81, 135, 405. Each factor divides 405 without leaving a remainder. - Every factor of a number is less than or equal to the number, eg. 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 are all less than or equal to 120 and 1, 3, 5, 9, 15, 27, 45, 81, 135, 405 are all less than or equal to 405.
Steps to find Factors of 120 and 405
- Step 1. Find all the numbers that would divide 120 and 405 without leaving any remainder. Starting with the number 1 upto 60 (half of 120) and 1 upto 202 (half of 405). The number 1 and the number itself are always factors of the given number.
120 ÷ 1 : Remainder = 0
405 ÷ 1 : Remainder = 0
120 ÷ 2 : Remainder = 0
405 ÷ 3 : Remainder = 0
120 ÷ 3 : Remainder = 0
405 ÷ 5 : Remainder = 0
120 ÷ 4 : Remainder = 0
405 ÷ 9 : Remainder = 0
120 ÷ 5 : Remainder = 0
405 ÷ 15 : Remainder = 0
120 ÷ 6 : Remainder = 0
405 ÷ 27 : Remainder = 0
120 ÷ 8 : Remainder = 0
405 ÷ 45 : Remainder = 0
120 ÷ 10 : Remainder = 0
405 ÷ 81 : Remainder = 0
120 ÷ 12 : Remainder = 0
405 ÷ 135 : Remainder = 0
120 ÷ 15 : Remainder = 0
405 ÷ 405 : Remainder = 0
120 ÷ 120 : Remainder = 0
Hence, Factors of
120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120
And, Factors of
405 are 1, 3, 5, 9, 15, 27, 45, 81, 135, and 405
Examples of GCF
Sammy baked 120 chocolate cookies and 405 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 120 and 405.
GCF of 120 and 405 is 15.
A class has 120 boys and 405 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 120 and 405. Hence, GCF of 120 and 405 is 15.
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 120 cans of Pepsi and 405 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 120 and 405. Hence GCF of 120 and 405 is 15. So the number of tables that can be arranged is 15.
Ariel is making ready to eat meals to share with friends. She has 120 bottles of water and 405 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 120 and 405. So the GCF of 120 and 405 is 15.
Mary has 120 blue buttons and 405 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 120 and 405. Hence, the GCF of 120 and 405 or the greatest arrangement is 15.
Kamal is making identical balloon arrangements for a party. He has 120 maroon balloons, and 405 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 120 and 405. So the GCF of 120 and 405 is 15.
Kunal is making baskets full of nuts and dried fruits. He has 120 bags of nuts and 405 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 120 and 405. So the GCF of 120 and 405 is 15.
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 120 bus tickets and 405 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 120 and 405. Hence, GCF of 120 and 405 is 15.