State two properties of H.C.F by division method
properties of HCF, or properties of the method?
The division method is one of the methods used to find the highest common factor (H.C.F) of two or more numbers. Here are two properties of H.C.F by the division method:
1. The H.C.F obtained by the division method is always a common factor of the given numbers. When we divide the given numbers by their H.C.F using the division method, the remainder will be zero for each division. This indicates that the H.C.F divides both numbers without leaving a remainder.
2. The H.C.F obtained by the division method is always greater than or equal to the common factors of the given numbers. It is the greatest common factor among all the common factors of the given numbers. This means that there is no other common factor that is greater than the H.C.F obtained by the division method.
To determine the Highest Common Factor (H.C.F) using the division method, two important properties can be observed:
1. The H.C.F of any two numbers divides their difference:
- In this method, numbers are divided successively until a remainder of zero is obtained. The last non-zero remainder is the H.C.F.
- If the difference of two numbers is divisible by a number, then that number is also a factor of both original numbers.
- For example, if we want to find the H.C.F. between 24 and 16, we divide 24 by 16, which gives a remainder of 8. If we divide the difference (24-16) = 8 by the remainder (8), we get a quotient of 1 with no remainder, indicating that 8 is the H.C.F. This property holds true for other numbers as well.
2. The H.C.F of any two numbers divides their product divided by their H.C.F:
- Once the H.C.F is obtained using division, it can be used to verify another property.
- If two numbers (a and b) have a common factor x, then both numbers can be written as a = x * p and b = x * q, where p and q are positive integers.
- The product of a and b is a * b = (x * p) * (x * q) = x^2 * p * q.
- The H.C.F. (x) also divides the product (x^2 * p * q) without leaving a remainder. Thus, the H.C.F divides the product of two numbers divided by their H.C.F.
These properties can be useful when finding the H.C.F of numbers using the division method.