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A number can be expressed as a product of its prime factors by prime factorization. A prime number has only two factors: one and the number itself. Consider the number 30 as an example. Although 30 = 5× 6 is a prime number, 6 is not. The number 6 can also be factored as 2 ×3 (prime numbers 2 and 3). As a result, 30 has a prime factorization of 2× 3× 5 with all factors being prime numbers.
Let’s learn more about prime factorization by working through a variety of mathematics problems, which will be followed by solved examples and practice questions.
Meaning of Prime Factorization
Prime factorization is the technique of expressing a number as the sum of prime numbers. The only two elements in prime numbers are 1 and the number itself. Prime numbers include numbers such as 2, 3, 5, 7, 11, 13, 17, 19, and so on. Any number can be represented as a product of prime numbers when it is prime factored.
Prime Factorization using Division Method
By dividing a huge integer by prime numbers, you can get the prime factors. To find the prime factors of an integer using the division method, follow these steps:
Step 1: Divide the number by the smallest prime number in such a way that the smallest prime number entirely divides the number.
Step 2: Divide step 1’s quotient by the smallest prime number once more.
Step 3: Repeat steps 2 and 3 until the quotient is 1.
Step 4: Finally, multiply all the divisors’ prime factors.
Let’s use the division approach to prime factorise 60.
Prime factorization of 60 = 2 × 2 × 3 × 5
Hence, the prime factors of 60 are 2, 3, and 5.
HCF and LCM Tricks
Two or more numbers have an H.C.F that is less than or equal to the smallest number of supplied numbers.
L.C.M of a, b, and c is the lowest number precisely divisible by a, b, and c.
The greatest number of provided numbers is larger than or equal to the L.C.M. of two or more numbers.
In each example, the smallest number that leaves a remainder R when divided by a, b, and c. Required number = (L.C.M of a, b, c) + R
The greatest number which divides a, b and c to leave the remainder R is H.C.F of (a – R), (b – R) and (c – R)
The greatest number which divide x, y, z to leave remainders a, b, c is H.C.F of (x – a), (y – b) and (z – c)
The smallest number which when divided by x, y and z leaves remainder of a, b, c (x – a), (y – b), (z – c) are multiples of M
Required number = (L.C.M of x, y and z) – M
Conclusion
The term “Least Common Multiple” is abbreviated as LCM. The smallest number that may be divided by both numbers is called the least common multiple of two numbers. It can be done with two or more integers or fractions.
To find the LCM of two numbers, there are several techniques. The product of the highest powers of the common prime factors is the LCM of those numbers, which is one of the easiest ways to find it. In mathematics, the least common multiple (or LCM) is referred to as the lowest common multiple. The smallest number among all possible multiples of two or more numbers is the least frequent multiple.
The largest possible number that splits both integers exactly is the highest common factor (HCF).
HCF represents the HCF of a and b. (a, b).
If d is the HCF (a, b), the common factor of a and b cannot be bigger than d.
The largest common divisor is another name for the highest common factor (HCF) (GCD)
The product of the least powers of the common prime factors is the HCF of those numbers, which is one of the easiest ways to find it.
