Factors
Category : 5th Class
Factors
Factors of a number, divide the number completely.
If a, b, c, d __ are factors of "m" then 'm will be completely divisible by a, b, c, d__.
How to Get Factors of a Number
Factors of a number can be found by hit and trial method. Get any number, if it divides completely the number whose factor is to be found, it is a factor of that number. Let us discuss some rules of divisibility in order to easily find the factors of a number.
Rules of divisibility
(a) The numbers which have 0, 2, 4, 6, or 8 at the unit place is divisible by 2. For example: 24434, 21450, 231545452218 are divisible by 2.
(c) If sum of digits of a number is divisible by 3 then the number is divisible by 3. For example: Sum of the digits of 276 = 2 + 7 + 6 = 15. 15 is divisible by 3, therefore, 276 is divisible by 3.
(d) If the number formed by two digits from right side of a number is divisible by 4 the number is divisible by 4. For example: 28 in 5428 is divisible by 4, therefore, 5428 is divisible by 4
(e) If a number has the digit 0 or 5 at unit place, the number is divisible by 5. For example: 0 is at the unit place in the number 5450, therefore, .5450 is divisible by 5.
(f) If an even number is divisible by 3 then the number is divisible by 6. For example: 558 is an even number and divisible by 3, therefore, 558 is divisible by 6
(g) If the number formed by three digits from right side of a number is divisible by 8 then the number is divisible by 8. For example: 248 in 56248 is divisible by 8, thus 56248 is divisible by 8.
(h) If sum of digits of a number is divisible by 9, the number is divisible by 9. For example: Sum of digits of 5689485 = 5 + 6 + 8 + 9 + 4 + 8 + 5 = 45 and 45 is the divisible by 9. Thus 9689485 is divisible by 9.
(i) If a number has the digit 0 at the unit place, the number is divisible by 10. For example: 0 is at the unit place in the number 4560, 4560 is divisible by 10.
(j) If difference of the sum of the alternate digits of a number is either 0 or divisible by 11, the number is divisible by 11. For example: Difference of the sum of the alternative digits of 5478693 = (5+7+6+3)-(4+8+9) = 0 Thus 5478693 is divisible by 11.
(k) If a number has two prime factors then product of the prime factors is also a factor of the number. For example: 3 and 5 are the prime-factors 2445 thus 15 is also a factor of 2445.
Find the factors of 10.
Answer:
By hit and trial method we get the numbers 1, 2, 5, and 10 which divide 10 completely.
How many factors are there of 56?
Answer:
Factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56. So there are 8 factors of 56.
Is 8 a factor of 45684?
Answer:
The number formed by the three digits from right side is 684 and 684 is not divisible by 8. Therefore, 8 is not a factor of 45684.
What least number should be subtracted from 16639 so that 24 becomes factor of it?
Solution:
Divide 16639 by 24, the remainder you will get, should be subtracted from 16639 so that 24 becomes factor of it.
\[24)\frac{693}{\begin{align} & 16639 \\ & \frac{144}{0223} \\ & \frac{-216}{0079} \\ & \frac{-72\,\,\,}{07} \\ \end{align}}(\]
Prime Numbers
The numbers which have only two factors, 1 and the number itself are called prime numbers.
Factors of 2 = 1, 2
Factors of 3 = 1, 3
Factors of 5 = 1, 5
Factors of 19 = 1, 19
We see all the above numbers 2, 3, 5, and 19 has only two factors 1 and the number itself. Therefore, all the above numbers are prime numbers.
Twin Primes
Two consecutive prime numbers with the difference 2 are called twin primes.
Some pairs of twin primes are the following:
Pairs of twin primes: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31)
Prime Triplets
Prime triplet is a set of three prime numbers which consists of a pair of twin primes and one other prime number which differs from one of the numbers of twin primes by 4 and 6 from other. There are two forms of prime triplet (p, p + 2, p + 6) and (p, p + 4, p + 6) where p is a prime number. In the (p, p + 2, p + 6) form of prime triplet p and p + 2 is a pair of twin primes. In the (p, p + 4, p + 6) form of prime triplet p + 4 and p + 6 is a pair of twin primes. (2, 3, 5) and (3, 5, 7) are two exceptions of prime triplet.
Some sets of prime triplet are the following:
(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47)
Composite Numbers
A number which has more than two factors is called a composite number.
Factors of 4 = 1, 2, 4
Factors of 6 = 1, 2, 3, 6
Factors of 9 = 1, 3, 9
All the above numbers 4, 6, and 9 have more than two factors. Therefore, these are composite numbers.
Perfect Numbers
If sum of all the factors of a number is twice of the number, the number is called a perfect number.
Factors of 6 = 1, 2, 3, 6
Sum of factors =1+2+3+6= 12
Sum of factors = 2 x the number. Therefore, 6 is a perfect number.
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