6th Class Mathematics Fractions LCM (Least Common Multiple)

  LCM (Least Common Multiple)

In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b usually is denoted by LCM (a, b). LCM is the smallest positive integer which is a multiple of both a and b.

The LCM of 12 and 80 is obtained by the product of greatest power of the prime factors. Therefore, the prime factors of \[12=2\times 2\times 3\]and prime factors of \[80=2\times 2\times 2\times 2\times 5=80.\] Now, LCM is obtained by multiplying factors those having greatest power occurred in either numbers. 12 has one 3, and 80 has four 2's and one 5, some multiply 2 four times, 3 once and five once. This gives us 240, the smallest number that can be exactly divided by both 12 and 80. Therefore, the LCM of two or more numbers is obtained by its prime factorization and common division method.  

LCM by Prime Factorization Method

LCM of two or more numbers is obtained by the following:

Step 1:   Prime factorization of every number.

Step 2:   Product of highest power of prime factors is the LCM of the numbers.  

Find the LCM of 24 and 46 by prime factorization method.

(a) 564                                                 

(b) 546

(c) 552                                                  

(d) All of these

(e) None of these             

Answer: (c)                

                 Prime factors of 24 \[=2\times 2\times 2\times 3=24\]

Prime factors of 46 \[=2\times 23=46\]

In prime factors of 24, 2 occurred three times but in prime factors of 46,2occurred only one time, therefore, the maximum power of 2 is 3 .  

Factor 3 is only one time in the prime factors of 24 but in the prime factors of 46, 3 is not a prime factor therefore, it has maximum power 1.

23 is also a prime factor of 46 and it has maximum power 1. Hence, LCM of 24 and \[46=2\times 2\times 2\times 3\times 23=552.\]Therefore, option (c)is correct and rest of the options is incorrect.  

LCM by Division Method

The following steps are used to determine the LCM of two or more numbers by division method:

Step 1:   Numbers are arranged or separated in a row by commas.

Step 2:   Find the number which divides exactly at feast two of the given numbers.

Step 3 :  Follow step 2 tiff there are no (at (east two) numbers divisible by any number.

Step 4:   LCM is the product of all divisor and indivisible numbers.  

The LCM of 15,45 and 65  \[=3\times 5\times 3\times 13=585\]

In the above division, 3 and 5 are divisor and 3 and 13 are indivisible numbers. Therefore, LCM is the product of divisors and indivisible numbers.  

Find the LCM of 8, 24, 28 by division method.

(a) 164                                                 

(b) 168

(c) 52                                                    

(d) All of these

(e) None of these  

Answer: (b)                

Explanation                 

 \[=2\times 2\times 2\times 3\times 7=168\]

Applications of LCM  

The least number which is exactly divisible by 120, 130 and 160 is.

(a) 6250                                               

(b) 6245

(c) 6240                                               

(d) All of these

(e) None of these  

Answer: (c)                

Explanation                

 \[=2\times 2\times 2\times 5\times 3\times 13\times 4=6240\]

Find the nearest numbers to 5000 which Is exactly divisible by each of 2, 3, 4 5, 6, 7, 8, and 9.

(a) 2520,5040                                    

(b) 2530,5050

(c) 2540,5060                                     

(d) All of these

(e) None of these  

Answer: (a)  

Explanation

The nearest number to 5000 which is exactly divisible by each one of the given numbers = 5000 - {Remainder of division 5000 - LCM of all divisors} - 5000 - {Remainder of division 5000 - 2520} = 5000 - 2480 = 2520 Another Nearest number to which is exactly divisible by each one of the = 5000 + {2520 - 3480} = 5000 + 40 = 5040.