Find the least number which when divided by \[6\] , \[15\] and \[18\] , leaves the remainder \[5\] in each case.? \[18\]

Question

Find the least number which when divided by  $$6$$ ,  $$15$$  and  $$18$$ , leaves the remainder  $$5$$  in each case.?  $$18$$

Vedantu Content Team · Accepted Answer

Hint: In this problem we have to find the least number which leaves as remainder  $$5$$  when divided by  $$6$$ ,  $$15$$  and  $$18$$ . We will first find the LCM of  $$6$$ ,  $$15$$  and  $$18$$  . so we can get the least number that gives zero as remainder when performing division . Now we add  $$5$$  to obtain LCM to get the required least number.  To find the LCM of a given number we can use multiple methods. Complete step-by-step answer:This problem is based on the application of LCM. The Lowest Common Multiple of two or more numbers is the smallest number which is the common multiple of those two or more numbers.  For example,  the LCM of   $$3$$  and  $$2$$  is  $$6$$ . i.e.    $$6$$  is the smallest number which is multiple of both  $$3$$  and  $$2$$   . Similarly  $$45$$  is LCM of  $$5$$  and   $$9$$ . We can either use the long division method or common multiples method or prime factorization method to find the LCM of any given number. Consider the given question,  The given numbers are $$6$$ ,  $$15$$  and  . Firstly we will multiply these numbers.  Multiples of    $$6$$  are:   $$6$$ , $$12$$ ,  $$18$$  , $$24$$  , $$30$$  , $$36$$  , $$42$$  , $$48$$  , $$54$$  , $$60$$  , $$66$$  , $$72$$  , $$78$$  , $$84$$  , $$90$$  , $$96$$  …….Multiples of  $$15$$  are:  $$15$$  ,  $$30$$   ,  $$45$$  , $$60$$  , $$75$$ ,  $$90$$  , $$105$$ ………Multiples of  $$18$$  are:  $$18$$ ,  $$36$$ ,  $$54$$ ,  $$72$$ ,  $$90$$  ,  $$108$$ ……..Lowest common multiple of  $$6$$ ,  $$15$$  and  $$18$$  is  $$90$$ Hence,  the LCM of  $$6$$ ,  $$15$$  and  $$18$$  is  $$90$$ .Now we have to find the least number which when divided by  $$6$$ ,  $$15$$  and  $$18$$  gives  $$5$$  as a remainder.  For this we add  $$5$$  to LCM of  $$6$$ ,  $$15$$  and  $$18$$   Hence we get,  $$(90 + 5) = 95$$   Hence,  $$95$$  is the smallest number which when divided by  $$6$$ ,  $$15$$  and  $$18$$  gives   $$5$$   as a reminder. Note: LCM of two or more numbers is always greater than or equal to those two or more numbers.  LCM  $$ \times $$  HCF   $$ = $$  Product of Two numberHCF is the highest common factor. LCM of two prime numbers is always a product of primes.  For example,  LCM of   $$3$$  and   $$2$$  is  $$6$$ .\t ( i.e. product of primes).