**Category : **5th Class

When two or more than two numbers are multiplied with each other, the resulting number is the multiple of all that numbers. For example, if A x B = C, C is multiple of both A and B.

**Multiples of 5 = 5, 10, 15, 20, 25, ____ **

**Multiples of 2 = 2, 4, 6, 8, 10, 12,____ **

**Multiples of 10 = 10, 20, 30, 40, ____**

Common Multiples

The same multiples of two or more than two different numbers are called common multiples.

** Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40,__ **

**Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45,___**

Common multiples of 4 and 5 = 20, 40, 60, etc.

Least Common Multiple (L.C.M.) The least common multiple among the common multiples of two or more than two numbers is called least common multiple.

**Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ___ **

**Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ___**

Common multiples of 5 and 7 = 35, 70,___

Least common multiple of 5 and 7 = 35.

**LCM by Prime Factorization Method**

In the prime factorization method numbers are written in the form of product of prime factors. The common and non-common factors are separated out and multiplied with each other. Their product are the LCM of the given numbers.

**Prime factorization of \[28=2\times 2\times 7\] **

**Prime factorization of \[36=2\times 2\times 3\times 3\] **

**Common prime factors \[=2\times 2\] **

**Non-common factors \[=7\times 3\times 3\]**

Product of the common factors \[=2\times 2\times 3\times 3\times 7=252\]

Thus LCM of 28 and 36 = 252.

**LCM by Division Method**

Division method of LCM involves the following steps:

** Step 1:** Write the numbers whose LCM is to be found in a row.

**Step 2:** Choose the least prime number by which at least two numbers are divisible. Then divide the numbers by the prime number.

**Step 3:** Write the quotient and undivided number in the next row just below the respected number.

**Step 4:** Repeat the process of division by the prime numbers unless only 1 or co-prime numbers remain in the last row.

**Step 5:** Now multiply the divisor and co-prime numbers of the last row. Product is the LCM.

**Find the LCM of 12, 20, 32 and 28**.

2 | 12, 20, 32, 28 |

2 | 6, 10, 16, 14 |

3 , 5, 8, 7 |

\[2\times 2\times 3\times 5\times 8\times 7=3360.\]

**Relation between Two Numbers and Their LCM and HCF**

(a) \[LCM\times H.C.F=\]Product of the two numbers

(b)\[\text{LCM=}\frac{\text{Product of the numbers}}{\text{H}\text{.C}\text{.F}}\]

(c) \[\text{HCF=}\frac{\text{Product of the numbers}}{\text{L}\text{.C}\text{.M}}\]

(d) Required number\[\text{=}\frac{\text{L}\text{.C}\text{.M }\!\!\times\!\!\text{ H}\text{.C}\text{.F}}{\text{Given}\,\text{number}}\]

**Find the LCM and HCF of 40 and 45 and verify that product of 40 and 45 is equal to the product of their LCM and HCF.**

LCM of 40 and 45 =360

HCF of 40 and 45 =5

Product of the numbers \[=40\times 45=1800\]

Product of their HCF and \[\text{LCM}=360\times 5=1800.\]

**If HCF and LCM of the two numbers are 10 and 300 respectively and one of the numbers is 50 find the other number**.

**Solution:**

Required number\[\text{=}\frac{\text{L}\text{.C}\text{.M }\!\!\times\!\!\text{ H}\text{.C}\text{.F}}{\text{Given}\,\text{number}}\] \[\text{=}\frac{\text{10}\,\text{ }\!\!\times\!\!\text{ }\,\text{300}}{50}\] = 60.

- Every number is a factor and a multiple of itself.
- Product of two numbers is a multiple of each of the number.
- Every number is a multiple of 1.
- Greatest multiple of a number can net be found.

- Factors of a number divide the number completely.
- Prime numbers have only two factors.
- Composite numbers have more than two factors.
- If two prime numbers differ by 2, these are twin primes.
- Prime triplet contains a pair of twin prime.
- Sum of the factors of a perfect number is twice of the number.
- Multiple of a number is the product of the number and a natural number.
- Product of the two numbers = Product of their LCM and HCF.

**Which one of following sets contains all the factors of 72?**

(a) {1,2, 4, 8, 9, 6, 12, 36, 72}

(b) {1,2, 3, 4, 6, 8, 9, 12, 18, 24, 48, 72}

(c) {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}

(d) {2, 3, 4, 6, 8, 9, 12,18, 24, 36, 48, 72}

(d) None of these

**Answer: (c)**

**Explanation**

Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36,and 72

**When a number is divided by its factor the remainder will be__.**

(a) 1

(b) 10

(c) 0

(d) 9

**Answer: (c)**

**Explanation**

A number is completely divisible by its factors leaving no remainder.

**Which one of the following digits should be placed in the middle of the digits of the number 258970 so that 3 becomes factor of it?**

(a) 0

(b) 1

(c) 2

(d) 3

(e) None of these

**Answer: (c)**

**Explanation**

Sum of digits of the number \[258970=2+5+8+9+7+0=31.\]therefore, 2 should be placed in the middle of the digits such that sum of the digits becomes divisible by 3.

**You have to add a least number in 1025629 in order to 12 be a factor of it. Choose the number from the following options and add.**

(a) 10

(b) 11

(c) 12

(d) 8

(e) None of these

**Answer: (b)**

**Which one of the following statements is not true about the factor?**

(a) The greatest factor of a number is the number itself

(b) The smallest even number is a factor of all the even numbers

(c) If X and Y are the factors of each other then X = Y

(d) X and Y are two natural numbers. A is a factor of X and B is a factor of Y If X > Y Then A > B

(e) None of these

**Answer: (d)**

**17, 19 is a pair of twin prime. To make it a prime triplet which one of the following prime numbers would you like to choose?**

(a)11

(b) 13

(c) 17

(d) 19

(e) None of these

**Answer: (b)**

**Explanation**

17-13 = 4 ,

19 -13 = 6. Thus 13 differs from one of the numbers of twin prime by 4 and 6 from other;

**If X is a composite number then what cannot be correct about X?**

(a) Y is a prime number and a factor of X

(b) X is divisible by 2

(c) Sum of the smallest and the greatest factors of X is 1 more than X

(d) If X lies between 10 and 20, it has no prime factor

(e) None of these

**Answer: (d)**

**Explanation**

12,14,15,16,18, are the composite numbers which lie between 10 and 20. 12, 14, 16, 18 are divisible by 2 which is a prime number. Thus X has the prime factors even if it lies between 1 and 20.

**Which one of the following is a perfect number?**

(a) 8

(b) 10

(c) 14

(d) 28

(e) None of these

**Answer: (d)**

**Q is a prime number and a member of a prime triplet. If Q = 17, which one of the following can not be a member of the prime triplet which contains Q?**

(a) 11

(b) 13

(c) 19

(d) 29

(e) 23

**Answer: (d)**

**Which one of the following is correct explanation for a prime number not to be a perfect number?**

(a) Prime numbers and perfect numbers are two different numbers therefore, a prime number cannot be perfect number

(b) Only a composite number can be a perfect number because a prime number has only two factors

(c) A prime number cannot be a perfect number because it has only two factors 1 and the number itself and sum of these two factors cannot be twice of the number

(d) If sum of the factors of a number except the number is equal to the number then it is a perfect number

(e) None of these

**Answer: (c)**

**Find the HCF of 256 and 400.**

(a) 4

(b) 8

(c) 16

(d) 32

(e) None of these

**Answer:(c)**

**Jack:** HCF of two numbers remains same even if the numbers are multiplied by any number.

**Codi:** HCF of two numbers gets doubled when the numbers are doubled. Who is correct?

(a) Jack

(b) Codi

(c) Both are correct

(d) Both partially incorrect

(e) None of these

**Answer: (b)**

**Explanation**

Codi is correct because when two numbers are multiplied by a same number then their HCF also increase as the product of the original HCF and the number

**Which one of the following statements is not true?**

(a) If X and Y are the two co-prime numbers, their HCF is 1

(b) If HCF of X and Y is X then Y is divisible by X.

(c) If HCF of X and Y is equal to the HCF of Y and Z, HCF of X and Z is also the same

(d) lf X + l = Y, X and Y are co-prime numbers

(e) None of these

**Answer: (c)**

**If Y = 2X, Z = 5Y, HCF of XYZ =:**

(a) X

(b) Y

(c) Z

(d) All of these

(e) None of these

** Answer: (a)**

**Sam chooses two numbers X and Y and he performs the long division method by dividing Y by X. He notices the last divisor is 1. Which one of the following is not true?**

(a) X and Y have only one common factor

(b) X and Y are the co-prime numbers

(c) HCF of X and Y is 1

(d) X and Y are the even numbers

(e) None of these

**Answer: (d)**

**Which one of the following is the LCM of 30 and 45?**

(a) 45

(b) 90

(c) 180

(d) 360

(e) None of these

**Answer: (b)**

** X is a 5 digit prime number. Sum of the digits of X is 28. You have to make it a multiple of least odd prime number by putting one more digit in the right side of the number. Which one of the following digits would you like to put?**

(a) 0

(b) 1

(c) 2

(d) 3

(e) None of these

**Answer: (c)**

**Explanation**

Sum of the digits of X is 28 and the least odd prime number is 3.Jhus you should put 2 in the right side of the number so that sum of digits of the number be divisible by 3.

**What two digit least number should be added to 25427 so that it becomes multiple of 25?**

(a) 23

(b) 21

(c) 25

(d) 27

(e) None of these

**Answer: (a)**

**X is a multiple of Y, Y is a multiple of Z, and Z is a multiple A and x > y. Which one of the following is the least common multiple of X, Y, Z, A?**

(a) X

(b) Y

(c) Z

(d) A

(e) None of these

**Answer: (a)**

**If LCM of two numbers is 600 and the numbers are 150 and 200 then find their HCF.**

(a) 50

(b) 60

(c) 75

(d) 80

(e) None of these

**Answer: (a)**

**Explanation**

\[\text{HCF=}\frac{\text{Product of Numbers }\!\!~\!\!\text{ }}{L.C.M}=\frac{150\times 200}{600}=50\]

So the HCF = 50.

** HCF and LCM of two numbers are 4 and 48 respectively. If one of the numbers is 2 more than X (Where X = 10) then find the other number.**

(a) 10

(b) 12

(c) 16

(d) 18

(e) None of these

**Answer: (c)**

**Explanation**

Given X = 10 Therefore, one of the numbers = 10 + 2 = 12 Required number \[=\frac{4\times 48}{12}=16.\]

**Jack:** If X is a LCM of two co-prime numbers, X is equal to the product of the prime numbers.

**Codi:** HCF of two or more than two numbers is always a factor of their LCM. Who is correct?

(a) Jack

(b) Codi

(c) Both are correct

(d) Both partially incorrect

(e) None of these

**Answer: (c)**

**Which one of the following statements is not correct?**

(a) If X is a natural number and Y is a multiple of X then Y is the LCM of X and Y

(b) lf X = Y, LCM of X and Y = HCF of X and Y

(c) A number is the smallest multiple and greatest factor of itself

(d) LCM of two prime numbers is a prime number

(d) None of these

**Answer: (d)**

**Find the least number when divided by 20, 25 and 50 leaves remainder 9 in each case?**

(a) 89

(b) 109

(c) 209

(d) 229

(e) None of these

**Answer: (b)**

*play_arrow*Factors and Multiples*play_arrow*Introduction*play_arrow*Factors*play_arrow*Common Factors*play_arrow*Multiples*play_arrow*Factors and Multiples*play_arrow*Factors and Multiples

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