Factors and Multple

Concepts

Divisibility Rules with Examples

1. Divisibility by 2

A number is divisible by 2 if its units (rightmost) digit is an even number: 0, 2, 4, 6, or 8.

Example:

  • 124 is divisible by 2 because the units digit is 4.
  • 137 is not divisible by 2 because the units digit is 7.

124 ÷ 2 = 62 (Divisible by 2)
137 ÷ 2 = 68.5 (Not divisible by 2)

2. Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Example:

  • For 123: 1 + 2 + 3 = 6, which is divisible by 3.
  • For 124: 1 + 2 + 4 = 7, which is not divisible by 3.

123 ÷ 3 = 41 (Divisible by 3)
124 ÷ 3 = 41.33 (Not divisible by 3)

3. Divisibility by 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4 or if it ends with two zeros.

Example:

  • For 132: The last two digits are 32, and 32 ÷ 4 = 8.
  • For 1500: It ends with two zeros, so it is divisible by 4.

132 ÷ 4 = 33 (Divisible by 4)
1500 ÷ 4 = 375 (Divisible by 4)

4. Divisibility by 5

A number is divisible by 5 if its units digit is 0 or 5.

Example:

  • For 105: The units digit is 5.
  • For 220: The units digit is 0.

105 ÷ 5 = 21 (Divisible by 5)
220 ÷ 5 = 44 (Divisible by 5)

5. Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3.

Example:

  • For 54: It is even (divisible by 2) and the sum of digits (5 + 4 = 9) is divisible by 3.
  • For 55: It is not even, so it is not divisible by 6.

54 ÷ 6 = 9 (Divisible by 6)
55 ÷ 6 = 9.17 (Not divisible by 6)

6. Divisibility by 7

A number is divisible by 7 if the difference between twice the units digit and the rest of the number is divisible by 7.

Example:

  • For 266: 26 – (2 × 6) = 14, and 14 is divisible by 7.
  • For 150: 15 – (2 × 0) = 15, which is not divisible by 7.

266 ÷ 7 = 38 (Divisible by 7)
150 ÷ 7 = 21.43 (Not divisible by 7)

7. Divisibility by 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8 or if it ends with three zeros.

Example:

  • For 1232: The last three digits are 232, and 232 ÷ 8 = 29.
  • For 4000: It ends with three zeros, so it is divisible by 8.

1232 ÷ 8 = 154 (Divisible by 8)
4000 ÷ 8 = 500 (Divisible by 8)

8. Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Example:

  • For 234: 2 + 3 + 4 = 9, which is divisible by 9.
  • For 245: 2 + 4 + 5 = 11, which is not divisible by 9.

234 ÷ 9 = 26 (Divisible by 9)
245 ÷ 9 = 27.22 (Not divisible by 9)

9. Divisibility by 10

A number is divisible by 10 if its units digit is 0.

Example:

  • For 130: The units digit is 0.
  • For 131: The units digit is 1.

130 ÷ 10 = 13 (Divisible by 10)
131 ÷ 10 = 13.1 (Not divisible by 10)

10. Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits in alternate places is 0 or divisible by 11.

Example:

  • For 3586: (3 + 8) – (5 + 6) = 11 – 11 = 0.
  • For 1234: (1 + 3) – (2 + 4) = 4 – 6 = -2, which is not divisible by 11.

3586 ÷ 11 = 326 (Divisible by 11)
1234 ÷ 11 = 112.18 (Not divisible by 11)

11. Divisibility by 12

A number is divisible by 12 if it is divisible by both 3 and 4.

Example:

  • For 144: It is divisible by 3 (1 + 4 + 4 = 9) and by 4 (last two digits 44 ÷ 4 = 11).
  • For 150: It is divisible by 3, but not by 4.

144 ÷ 12 = 12 (Divisible by 12)
150 ÷ 12 = 12.5 (Not divisible by 12)

12. Divisibility by 19

A number is divisible by 19 if the sum of the tens digit and twice the units digit is divisible by 19.

Example:

  • For 665: (66 + 2 × 5) = 66 + 10 = 76, which is not divisible by 19.
  • For 114: (11 + 2 × 4) = 11 + 8 = 19, which is divisible by 19.

665 ÷ 19 = 35 (Not divisible by 19)
114 ÷ 19 = 6 (Divisible by 19)

Understanding Factors

A divisor of an integer n, also called a factor of n, is an integer that evenly divides n without leaving a remainder.

Key Points about Factors

  1. 1 (and -1) are divisors of every integer.
  2. Every integer is a divisor of itself.
  3. Every integer is a divisor of 0, except, by convention, 0 itself.
  4. Numbers divisible by 2 are called even, and numbers not divisible by 2 are called odd.
  5. A positive divisor of n which is different from n is called a proper divisor.
  6. An integer n > 1 whose only proper divisor is 1 is called a prime number.
    • A prime number has exactly two factors: 1 and itself.
    • Example: The number 5 is a prime number because its only factors are 1 and 5.
  7. Any positive divisor of n is a product of prime divisors of n raised to some power.
    • Example: The divisors of 12 can be expressed using its prime factorization: 2^2 \times 3^1.
  8. If a number equals the sum of its proper divisors, it is called a perfect number.
    • Example: The proper divisors of 6 are 1, 2, and 3. Since 1 + 2 + 3 = 6, 6 is a perfect number.

Examples

Below are examples of how to express factors and related concepts using LaTeX:



      \[ \text{Example 1: Factors of 12} \]


      \[ 12 = 2^2 \times 3 \]


      \[ \text{Proper divisors: } 1, 2, 3, 4, 6 \]



      \[ \text{Example 2: Prime Number} \]


      \[ \text{The number 7 is prime because its only divisors are 1 and 7.} \]



      \[ \text{Example 3: Perfect Number} \]


      \[ \text{The proper divisors of 28 are 1, 2, 4, 7, and 14:} \]


      \[ 1 + 2 + 4 + 7 + 14 = 28 \]


      \[ \text{Thus, 28 is a perfect number.} \]

Finding the Number of Factors of an Integer

To find the total number of factors of an integer, we first need to perform its prime factorization. Given an integer n expressed as:

    \[ n = a^p \times b^q \times c^r \]

where a, b, and c are prime factors of n, and p, q, and r are their respective powers.

The total number of factors of n can be calculated using the formula:

    \[ \text{Total number of factors} = (p + 1)(q + 1)(r + 1) \]

Note: This formula includes both 1 and the integer n itself as factors.

Example: Finding the Total Number of Factors of 450

Let’s find the prime factorization of 450 and use it to determine the number of factors.

  • Prime factorization of 450: 450 = 2^1 \times 3^2 \times 5^2
  • Applying the formula: (1 + 1)(2 + 1)(2 + 1)

    \[ \text{Total number of factors} = (1 + 1)(2 + 1)(2 + 1) = 2 \times 3 \times 3 = 18 \]

Therefore, 450 has a total of 18 factors.

List of All Factors of 450

To verify, let’s list all factors of 450:

Factors: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450

We can see that there are indeed 18 factors.

Another Example: Number of Factors of 60

  • Prime factorization of 60: 60 = 2^2 \times 3^1 \times 5^1
  • Applying the formula: (2 + 1)(1 + 1)(1 + 1)

    \[ \text{Total number of factors} = 3 \times 2 \times 2 = 12 \]

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Thus, 60 has a total of 12 factors.

Greatest Common Factor (GCF) or Greatest Common Divisor (GCD)

The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), of two integers is the largest integer that divides both numbers without leaving a remainder.

How to Find the GCF

  1. Perform the prime factorization of each number.
  2. Identify the common prime factors.
  3. Choose the lowest power of each common prime factor.
  4. Multiply these common factors to get the GCF.

Example 1: GCF of 36 and 48

  • Prime factorization of 36: 36 = 2^2 \times 3^2
  • Prime factorization of 48: 48 = 2^4 \times 3^1
  • Common factors: 2 and 3
  • Lowest power of each common factor: 2^2 and 3^1

    \[ \text{GCF} = 2^2 \times 3^1 = 4 \times 3 = 12 \]

Therefore, the GCF of 36 and 48 is 12.

Properties of GCF

  • Every common divisor of a and b is a divisor of \text{GCD}(a, b).
  • The product of two numbers is equal to the product of their GCD and LCM:

        \[ a \times b = \text{GCD}(a, b) \times \text{LCM}(a, b) \]

Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two integers is the smallest positive integer that is a multiple of both numbers.

How to Find the LCM

  1. Perform the prime factorization of each number.
  2. Identify all the prime factors.
  3. Choose the highest power of each prime factor.
  4. Multiply these factors to get the LCM.

Example 2: LCM of 36 and 48

  • Prime factorization of 36: 36 = 2^2 \times 3^2
  • Prime factorization of 48: 48 = 2^4 \times 3^1
  • Highest power of each prime factor: 2^4 and 3^2

    \[ \text{LCM} = 2^4 \times 3^2 = 16 \times 9 = 144 \]

Therefore, the LCM of 36 and 48 is 144.

Relationship Between GCF and LCM

The product of the GCF and LCM of two numbers is equal to the product of the numbers themselves:

    \[ \text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b \]

Example 3: Verification of GCF and LCM Relationship

  • For numbers 36 and 48:
  • GCF: 12
  • LCM: 144
  • Product of GCF and LCM: 12 \times 144 = 1728
  • Product of the numbers: 36 \times 48 = 1728

The relationship is verified: \text{GCD}(36, 48) \times \text{LCM}(36, 48) = 36 \times 48.

Class Materials

LCM, HCF, Factor, Multiple & Divisibility Test

One thing that will help you to find the prime factorization of a number is a quick way to tell if a number is divisible by smaller numbers.

[i] A counting number is divisible by 2 if its units digit is an even number. Thus, if the rightmost digit of a counting number is 0, 2, 4, 6, or 8, the number is divisible by 2.

[ii] A counting number is divisible by 3 if the sum of its digits is divisible by 3.

[iii] A counting number is divisible by 4 if the two-digit number formed by the tens and units digits is divisible by 4 or there is 2 terminating zero.

[iv] A counting number is divisible by 5 if its units digit is 0 or 5.

[v]A counting number is divisible by 6 if it is divisible by 2 and also by 3. Thus, if the units digit is even and the sum of all digits is divisible by 3, then the number is divisible by 6

[vi] A counting number divisible by 7 If difference of two times   The ones digits of the given number is divisible by 7. Example: 266, 26-2×6=14, 14 is divisible by 7. So,266 is divisible by 7

[vii] A counting number divisible by 8 If number formed by the last three digits of the given number is divisible by8 or there is three terminating zero then the given number is also divisible by 8

[viii] A counting number is divisible by 9 if the sum of its digits is divisible by 9.

[ix] A counting number is divisible by 10 if the units digit is 0.

[x] A counting number divisible by 11 If the difference of the sum of the digits at the alternate place is 0 or divisible by 11, then the given number is also divisible by 11. Example,3586,3+8=11,5+6=11 and, 11-11=0 so,3586 is divisible by 11.

[xi] A counting number divisible by 12 If the number is divisible by 3 and 4 both.

[xii]A counting number divisible by 19: IF the sum of number of tens in the number and twice the units digit is divisible by 19 then the given number is divisible by 19. Examples 665,969,873 456760

Factors

A divisor of an integer n, also called a factor of n, is an integer that evenly divides n without leaving a remainder.

[i] 1 (and -1) are divisors of every integer.

[ii] Every integer is a divisor of itself.

[iii] Every integer is a divisor of 0, except, by convention, 0 itself.

[iv] Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.

[v] A positive divisor of n which is different from n is called a proper divisor.

[vi] An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, one would say that a prime number is one which has exactly two factors: 1 and itself.

[vii]Any positive divisor of n is a product of prime divisors of n raised to some power.

[viii] If a number equals the sum of its proper divisors, it is said to be a perfect number.

Example: The proper divisors of 6 are 1, 2, and 3: 1+2+3=6, hence 6 is a perfect number.

Finding the Number of Factors of an Integer

First make prime factorization of an integer n = ap×bq×cr, where a, b, and c are prime factors of n and p, q, and r are their powers.

The number of factors of n will be expressed by the formula = [p +1][q +1][r+1] NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: 450 =213252

Total number of factors of 450 including 1 and 450 itself is [1+1][2+1][2+1] = 2×3×3 = 18  factors.

Greatest Common Factor (Divisor) – GCF (GCD)

To find the GCF, you will need to do prime factorization. Then, multiply the common factors (pick the lowest power of the common factors).

  • Every common divisor of a and b is a divisor of GCD (a, b).
  • a×b=GCD(a, b) ×lcm(a, b)

Lowest Common Multiple – LCM

To find the LCM, you will need to do prime factorization. Then multiply all the factors (pick the highest power of

the common factors).

PS Questions

1.  Each of the following numbers has a remainder of 2 when divided by 11 except:

A.2   

B.13   

C.24    

D.57    

E.185

2.  When the positive integer x is divided by 9, the remainder is 5. What is the remainder when 3x is divided by9?

A.0    

B.1   

C.3   

D.4       

E.6

3. If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1?

A.8  

B.9    

C.16     

D.23        

E.24

4. If k and x are positive integers and x is divisible by 6, which of the following CANNOT be the value of?

A.24k√3   

B.24√k     

C.24√(3k)

D.24√(6k)

E.72√k

5. x, y, a, and b are positive integers. When x is divided by y, the remainder is 6. When a is divided by b, the remainder is 9. Which of the following is NOT a possible value for y + b?

A.24    

B.21       

C.20              

D.17            

E.15

6. When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?

A.0

B.1

C.2

D.3

E.4.

7. A group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 5 with 3 students left over. What is the sum of the two smallest possible values of n?

A.33

B.46

C.49

D.53  

E.86

8. When x is divided by 4, the quotient is y and the remainder is 1. When x is divided by 7, the quotient is z and the remainder is 6. Which of the following is the value of y in terms of z?

A.(4z/7)+ 5

B.(7z + 5)/6

C.(6z + 7)/4

D.(7z + 5)/4

E.(4z + 6)/7

9. If n is an integer and n4 is divisible by 32, which of the following could be the remainder when n is divided by 32?

(A) 2

(B) 4

(C) 5

(D) 6

(E) 10

10. x1 and x2 are each positive integer. When x1 is divided by 3, the remainder is 1, and when x2 is divided by 12, the remainder is 4.

If y = 2x1 + x2, then what must be true about y?

I. y is even

II. y is odd

III. y is divisible by 3

(A) I only

(B) II only

(C) III only

(D) I and III only

(E) II and III only

12. If positive integer n is divisible by both 4 and 21, then n must be divisible by which of the following?

A.8

B.12

C.18   

D.24     

E.48

13. Which of the following is the lowest positive integer that is divisible by 8, 9, 10, 11, and 12?

A.7,920

B. 5,940

C.3,960  

D.2,970

E. 890

14. Which of the following is the lowest positive integer that is divisible by the first 7 positive integer multiples of 5?

A.140

B.210    

C.1400

D.2100  

E.3500

15. For any positive integer n, the length of n is defined as a number of prime factors whose product is n, for example, the length of 75 is 3, since 75=3x5x5. How many two-digit positive integers have a length 6?

A.0  

B.1  

C.2

D.3

E.4

16. If n is a non-negative integer such that 12n is a divisor of 3,176,793, what is the value of n12 – 12n ?

A. – 11

B. – 1

C.0    

D.1

E.11 

17. The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the value of n?

A.6  

B.8

C.9

D.12

E.15

17. How many factors does 362 have?

A.2

B.8

C.24

D.25  

E.26

18. a, b, and c are positive integers. If a, b, and c are assembled into the six-digit number abcabc, which one of the following must be a factor of abcabc?

(A) 16  

(B) 13

(C) 5    

(D) 3   

(E) none of the above

19. A restaurant pays a seafood distributor d dollar for 6 pounds of Maine lobster. Each pound can make v vats of lobster bisque, and each vat makes b bowls of lobster bisque. If the cost of the lobster per bowl is an integer, and if v and b are different prime integers, then which of the following is the smallest possible value of d?

(A) 15  

(B) 24   

(C) 36  

(D) 54  

(E) 90

Quantity Comparison

20.

The number of even factors of 27The number of even factors of 81

21.

The number of distinct factors of 10The number of distinct prime factors of 210

22.

The least common multiple of 22 and 6The greatest common factor of 66 and 99

23. m is a positive integer that has a factor of 8.

The remainder when m is divided by 6The remainder when m is divided by 12

24.

The remainder when 1011 is divided by 2The remainder when 313 is divided by 3

25. In set N consisting of n integers, the average equals the median.

The remainder when n is divided by 2The remainder when n – 1 is divided by 2

26. The product of integers f, g, and h is even and the product of integers f and g is odd

The remainder when f is divided by 2The remainder when h is divided by 2

Class Slide

Time-bound Tests

  • NP Primes Test – 1
  • NP Primes Test – 2
  • NP Primes Test – 3
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