Math Assignment Help With Divisibility Rules And Prime Numbers
Chapter 5. Divisibility Rules And Prime Numbers
5.1 Introduction: When a whole number is divisible by another number, remainder being zero, then the second number is called the factor of the first number. Divisibility is a method to determine whether the number is completely divisible by the other or not.
5.2 Divisibility Rules:
5.2.1 Divisibility by 2:
- All even number i.e. all the numbers having an even number at units place is divisible by 2
- Example: 0, 2, 4, 6, 8, 10, 12, 14, 22, 34, 56, etc.
5.2.2 Divisibility by 3:
- Add all the digits in a number.
- If the sum is divisible by 3, the number is divisible by 3
- Example: 345,
3+4+5 = 12
Since 12 is divisible by 3, the number 345 is also divisible by 3
5.2.3 Divisibility by 4:
- Check the last two digits of the number.
- If the last two digits are divisible by 4, the number is divisible by 4.
- Example: 4532
Last two digits are; 32
32/4 = 8
Remainder = 0
Hence the number 4532 is divisible by 4.
5.2.4 Divisibility by 5:
- All the numbers ending with digits 5 and 0 are divisible by 5
- Example: 375625
The last digit is 5, so the number is divisible by 5
5.2.5 Divisibility by 6:
- If the number is divisible by 2 and 3 both, the number is divisible by 6.
- Example: 72
The number is divisible by 2 as the last digit is even.
The number is divisible by 3 also as 7+2 = 9
And 9 is divisible by 3
Therefore the number 72 is divisible 6
5.2.6 Divisibility by 7:
- Multiply the last digit by 2
- Subtract the product from the rest of the number.
- If the result is divisible by 7, the number is divisible by 7.
- Example: 343
3 x 2 -= 6
34 – 6 = 28
28 is divisible by 7, so the number 343 is divisible by 7
5.2.7 Divisibility by 8:
- If the last three digits of a number is divisible by 8, the number is divisible by 8.
- Example: 3168
168/8 = 21
Therefore the number is divisible by 8
5.2.8 Divisibility by 9:
- Add all the digits in the number.
- If the sum is divisible by 9, the number is also is divisible by 9.
- Example: 342
3+4+2 = 9
Therefore the number is divisible by 9
5.2.9 Divisibility by 10:
- If the last digit of the number is 0 the number is divisible by 10
- Example: 10, 20, 30, 40, 100, and so on
5.2.10 Divisibility by 11:
- Add the alternate digits of the number.
- And the remaining set of alternate digits.
- If both the sums are equal, the number is divisible by 11.
5.3 Prime numbers:
5.3.1 Introduction: a prime number is a natural number which has only two divisors, 1 and the number itself.
2, 3,5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 and so on.
Prime numbers are whole numbers and are greater than1.
Twin prime numbers are the numbers that differ by 2.
(3,5), (5,7), (11,13)…. Etc.
5.3.2 Co-prime numbers: A pair of numbers not having any common factors other than 1 or -1 are called co-prime number.
Example: 11 and 25 are co-prime, because
the factors of 11 are 1 and 11
and the factors of 25 are 1, 5and 25
except 1 no other factor is common between them.
5.3.3Prime Factorization: Prime Factorization is a method of finding the prime numbers you need to multiply together to get the original number.
Example: What are the prime factors of 24?
Let’s start from the smallest prime number, that is 2,
24 / 2 = 12
But 12 is not a prime number, so factorize it further:
12/ 2 = 6
6 is also not a prime number, so factorize it further
6/2 = 3
3 is a prime number, so:
24 = 2 x 2 x2x 3
The prime factorization of 12 is 2 x 2 x2 x 3, and it can also be written as 23x 3
What is the prime factorization of 245?
We cannot divide 245 evenly by 2, so try the next prime number 3, this also not working so try the next one, i.e. 5:
245/ 5 = 49
Factorize 49, we find that 7 is the smallest prime number that works:
Now, 7 is a prime number.
So the prime factorization of 245 is 5 x 7 x 7 or 5 x 72
5.3.4 Least Common Multiple (L.C.M.): LCM of two natural numbers is the smallest natural number which is a multiple of both the numbers.
5.3.5 Highest Common Factor (H.C.F.): HCF of two natural numbers is the largest common factor (or divisor) of the given natural numbers. In other words, H.C.F. is the greatest element of the set of common factors of the given numbers.
H.C.F. is also called Greatest Common Divisor (G.C.D.)
5.3.6 Relation between L.C.M. and H.C.F. of two natural numbers
The product of L.C.M. and H.C.F. of two natural numbers = the product of the numbers.
Note.* In particular, if two natural numbers are co-prime then their L.C.M. = the product of the numbers.
5.4 General Divisibility Rule for any prime divisor 'p' :
Consider multiples of p. keep multiplying the prime divisor (n) by Multiple of p + 1 until the product reaches to the closest value of multiple of 10, so that one tenth of least multiple of p + 1 is a natural number.
Thus, n = one tenth of (least multiple of 'p' + 1).
Find (p - n) also.
Consider an Example:
Let the prime divisor be 3.
Multiples of 3 are 1x3, 2x3, 3x3
3 x3 = 9
9+1=10 is a multiple of 10
So 'n' for 3 is one tenth of (least multiple of p + 1) = (1/10)10 = 1
'p-n' = 3-3 = 0
Example (ii) :
Let the prime divisor be 11.
Multiples of 11 are 1x11, 2x11, 3 x11, 4x11, 5x11, 6x11, 7x11, 8x11, 9x11
99+1=100, is a multiple of 10
So 'n' for 11 is one tenth of (least multiple of 'p' + 1) = (1/10)100 = 10
'p-n' = 11 – 9= 2
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