Nice Skill development ppt from ethnus

2. NUMBER PROPERTIES- NUMBER OF FACTORS

3. Factors
Factors of a number are defined as numbers that divide the original
number evenly or exactly.
EXAMPLES:
• Factors of 2 = 1 and 2
• Factors of 10 = 1,2,5 and 10
• Factors of an algebraic expression are the values that can divide the given expression. For
example, factors of 3xyz are 3, x, y, z, 3x, 3y, 3z, xyz and 3xyz.

4. Factors
Properties of Factors
• All integers have a finite number of factors.
• A number’s factor is always less than or equal to the number; it can
never be bigger than the number.
• Except for 0 and 1, every integer has a minimum of two factors: 1 and
the number itself.
• Factors are found by employing division and multiplication.

5. Factors
Fun Facts about Factors
• Factors are never decimals or fractions; they are only whole numbers or
integers.
• All even numbers have 2 as a factor.
• 5 is a factor for all numbers that end in 0 and 5.
• All numbers higher than 0 and ending in a 0 have 2, 5, and 10 as
factors.
• Factoring is a common way to solve or simplify algebraic expressions.

6. Factors
Prime Factorization
When we write a number as a product of all its prime factors, it is called prime
factorization.
Every number in prime factorization is a prime number.
To write the number as a product of prime factors, sometimes we might have to
repeat the factors too.
Example: To write the prime factorization of 8, we can write 8 = 2 ✕ 2 ✕ 2, i.e, the
prime factor 2 is repeated three times.

7. Factors
Real-life Applications of Factorization
Equal division.
If six people come together to eat a whole pizza that has been cut into 24 slices, it would only be
fair that everyone receives an equal number of slices. Therefore, this pizza can be divided into
equal shares because 6 (the number of people) is a factor of 24 (the number of pizza slices).
When you divide 24 by 6, you get 4, and each individual receives four slices!
Factoring and money.
The exchange of money and its divisions into smaller units rely heavily on factoring. For
example, four quarters equal one dollar in America. In India, a rupee was further divided into 1
paisa, 5 paise, 10 paise, 25 paise, and 50 paise.

8. Factors
Important Formula
Number of factors
If we have to find the number of factors of any number say N, then we should follow below steps:
Step 1: Prime factorize N=p^a×q^b×r^c×…
Step 2: The number of factors of N= (a+1)(b+1)(c+1)…
Sum of factors
To find the sum of all the factors of a number (say N), we follow below two steps:
Step 1: Prime factorize N=p^a×q^b×r^c×…
Step 2: Sum of factors =[(p^(a+1)–1)/(p–1)][(q^(b+1)–1)/(q–1)][(r^(c+1)–1)/(r–1)]…

9. Factors
Important Formula
Product of Factors
To find the product of all the factors of a number (say N), we follow below three steps:
Step 1: Prime factorize N=p^a×q^b×r^c×…
Step 2: Let the number of factors of N be x. therefore, x= (a+1)(b+1)(c+1)…
Step 3: Product of factors =N^(x/2)

10. Question: 01
Find the number of factors of 240
A. 20
B. 9
C. 4
D. 12
Answer: A

11. Explanation:
Step 1: Get prime factors of a number say 240
240 = 2^4 * 3^1 * 5^1
Step 2: Number of factors of a number.
Number of factors = (4+1) * (1+1) * (1+1) = 5*2*2 =20
Thus the powers of the numbers are increased by one and multiplied.

12. Question: 02
Find the number of factors of 620 ?
A. 11
B. 12
C. 13
D. 14
Answer: B

13. Explanation:
Step 1: Get prime factors of a number say 620
620 = 2^2 * 5^1*31^1
Step 2: Number of factors of a number.
Number of factors = (2+1) * (1+1) * (1+1) = 3*2*2 =12
Thus the powers of the numbers are increased by one and multiplied

14. Question: 03
Find the number of even factors of 240?
A. 20
B. 16
C. 4
D. 8
Answer:B

15. Explanation:
Step 1: Get prime factors of a number say 240
240 = 2^4 * 3^1 * 5^1
Step 2: Number of Even factors formula is
240= (4) * (1+1)*(1+1) = 4*2*2 =16
Thus the powers of the numbers are increased by one and multiplied except 2.

16. Question: 04
Find the number of odd factors of 240?
A. 5
B. 4
C. 3
D. 2
Answer: B

17. Explanation:
Step 1: Get prime factors of a number say 240
240 = 2^4 * 3^1 * 5^1
Step 2: Number of odd factors formula is
240= (2^0) * (3^1)*(5^1) = 1*(1+1)*(1+1) =1*2*2 =4
Thus the powers of the numbers are increased by one and multiplied except 2.

18. Question: 05
Find the sum of factors of 240 ?
A. 450
B. 744
C. 480
D. 800
Answer: B

19. Explanation:
Calculate the sum of factors of a number:
Step 1: Get prime factors of a number say 240
240 = 2^4 * 3^1 * 5^1
Step 2: Sum of factors formula is
240= (2^0 + 2^1 + 2^2 + 2^3 + 2^4) * (3^0 + 3^1)*(5^0 + 5^1)
Step: 31*4*6 = 744

20. Question: 07
Find the number of Even and Odd factors of 1200.
A. 24,6
B. 6,24
C. 8,2
D. 2,8
Answer: A

21. Explanation:
Prime factorization of 1200=2^4×3^1×5^2
We know the fact that all the factors of 1200 will be in the form of 2^a×3^b×5^c.
For number of even factors = a×(b+1) ×(c+1)
therefore total number of even factors of 1200 = 4x(1+1)x(2+1)=24.
For number of odd factors = 2^0×(b+1) ×(c+1)
Hence the number of odd factors of 1200 = 1x2x3=6.

22. Question: 08
What is the product of all the factors of 180?
A. 1809.
B. 18018.
C. 1808.
D. 1807.
Answer: A

23. Explanation:
180= 223251. There are (2+1)(2+1)(1+1) or 18 factors.
If the given number is not a perfect square, at least one of the indices is odd and the number
of factors is even. We can form pairs such that the product of the two numbers in each pair is
the given number (180 in this example).
The required product is 1809.

24. Question: 09
Find the product of all the prime factors of 544.
A. 17
B. 34
C. 2
D. 1
Answer: B

25. Explanation:
Prime Factorization of 544 = 25 × 171
Since, the prime factors of 544 are 2, 17. Therefore, the product of prime factors = 2 × 17 =
34.

26. Question: 10
Find the Sum of odd factors of 240?
A. 24
B. 124
C. 90
D. 150
Answer: A

27. Explanation:
Step 1: Get prime factors of a number say 240
240 = 2^4 * 3^1 * 5^1
Step 2: Sum of Odd factors formula is
240= ( 2^0) * (3^0 + 3^1)*(5^0 + 5^1)
Step: 1*4*6 = 24

28. Question: 11
Find the number of factors which are perfect squares of 10800.
A. 12
B. 7
C. 6
D. 9
Answer: A

29. Explanation:
Prime factorization of 10800=2^4×3^3×5^2
If we prime factorise any number which is a perfect square, we would observe that in all cases
the exponent of all the prime factors of the number to be even only.
For example, 36 is perfect square 36=2^2×3^2. here we can see that the exponent of both 2
and 3 are even.
Again, any factor 10800 will be in the form of 2^a×3^b×5^c. For the factors to be perfect
squares, all the values a, b, and c has to be even only.
Or, the possible values which a can take for = 0, 2, 4, i.e. 3 values only. Similarly, b can take
0, 2 i.e. 2 values and c can take 0, 2 i.e. 2 values.
the number of factors which are perfect squares of 10800 = 2×2×3 = 12

30. Question: 12
In how many ways can 14630 be written as the product of two factors?
A. 15
B. 7
C. 12
D. 16
Answer: D

31. Explanation:
14630=2x5x7x11x19
Hence, 14630 can be written as the product of two factors in
½ {(1+1) (1+1) (1+1) (1+1) (1+1)} = 16 ways

32. Question: 13
In how many ways can the number 44100 can be written as a product of two different factors?
A. 40
B. 5
C. 36
D. 21
Answer: A

33. Explanation:
First expressing 44100 a product of its prime factors. We get 44100 = 22 x 32 x 52 x 72
Since all the powers are even, the given number is a perfect square. Since the question has
asked us to find the number of ways of writing the number as a product of two “different”
factors, we cannot consider (sq. root x sq. root).
So, the required number of ways is: ½ {(2+1) (2+1) (2+1) (2+1) -1} = ½ {81-1} =40.

34. Question: 14
How many factors of the number 28 * 36 * 54 * 105 are multiples of 120?
A. 540
B. 660
C. 594
D. 792
Answer: C

35. Explanation:
The prime factorization of 28 * 36 * 54 * 105 is 213 * 36 * 59.
For any of these factors questions, start with the prime factorization. Remember that the
formulae for number of factors, sum of factors, are all linked to prime factorization.
120 can be prime-factorized as 23 * 3 * 5.
All factors of 213 * 36 * 59 that can be written as multiples of 120 will be of the form 23 * 3 * 5 *
K.
213 * 36 * 59 = 23 * 3 * 5 * K
=> K = 210 * 35 * 58.
The number of factors of N that are multiples of 120 is identical to the number of factors of K.
Number of factors of K = (10 + 1) (5 + 1) * (8 + 1) = 11 * 6 * 9 = 594

36. Question: 15
Number N = 26 * 55 * 76 * 107; how many factors of N are even,odd numbers?
A. 1183
B. 1200
C. 1050
D. 540
Answer: A

37. Explanation:
The prime factorization of 26 * 55 * 76 * 107 is 213 * 512 * 76.
The total number of factors of N = 14 * 13 * 7
We need to find the total number of even factors. For this, let us find the total number of odd
factors and then subtract this from the total number of factors. Any odd factor will have to be a
combination of powers of only 5 and 7.
Total number of odd factors of 213 * 512 * 76 = (12 + 1) * (6 + 1) = 13 * 7=91
Total number of factors = (13 + 1) * (12 + 1) * (6 + 1)=1274
Total number of even factors = 14 * 13 * 7 - 13 * 7=1274-91
Number of even factors = 13 * 13 * 7 = 1183

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