Finding the Greatest n - digit Number which is exactly divisible.

 Finding the Greatest n - digit number when divided by the numbers leaves no remainder,using L.C.M :

B) i) Greatest n-digit number, which is exactly divisible by the numbers.

 Steps to find Greatest n - digit number which is divisible exactly by the numbers,(leaves no remainder):

1. Find out the L.C.M of the given numbers, ‘L’ .

2. Divide the largest n – digit number with the L.C.M( Largest n-digit ÷ L.C.M )  which leaves remainder,’R’.

3. Subtract the Remainder,’R’ from the largest n- digit number which gives the required Greatest n – digit number.

Required Largest n - digit number  = Largest n-digit number – Remainder'R' .   

 

1) Find the greatest number of 5 digits which is exactly divisible by 8, 9, 15, 21 is?

Solution:

Given : Numbers = 8, 9, 15, 21; Greatest 5 digit number = 99,999.

1.We need to find L.C.M of the given numbers: L.C.M(8, 9 15, 21 ) = 2520.

 

2. Divide the largest 5 digit number by L.C.M = 99999 ÷ 2520 , which leaves remainder ‘R’ = 1719.

3. Subtract the remainder ‘R’ from the largest 5 digit number which gives the required Greatest 5 digit number ,which is divisible exactly by the given numbers.  

99999 – 1719 = 98280, required greatest 5 digit number.

 

 

2) The greatest number of 4 digits which when divided by 2, 3, 4, 5, 6, and 7 is ?

Solution:

Given: Numbers = 2, 3, 4, 5, 6, 7 ; greatest 4 digit number = 9,999.

1. We need to find the L.C.M of the given numbers:

L.C.M( 2, 3, 4, 5, 6, 7 ) = 420.

2. Divide the largest 4 digit number with the L.C.M : 9999 ÷ 420 , which leaves remainder ‘R’ = 339 .

3. Subtract the obtained remainder ‘R’ from the largest 4 digit number,which gives the required Greatest 4 digit number, which is divisible exactly by the given numbers.

Required Greatest 4 digit number = Greatest 4 digit number - Remainder.

9999 – 339 = 9660, the required 4 digit greatest number.  

3) Find the greatest number of three digits which, when divided by 3, 4 and 5 leaves no remainder?

Solution:

Given: Numbers = 3, 4, 5 ; Greatest 3 digit number = 999.

1. Find the L.C.M of the given numbers: L.C.M (3, 4, 5) = 60.

As the given numbers are Consecutive numbers, the L.C.M of the numbers is the product of the given consecutive numbers.

Therefore, L.C.M( 3, 4, 5 ) = 3 * 4 * 5 = 60 = L.

2. Divide the greatest n-digit number,999 with L.C.M,’L’ 60 ; 999 ÷ 60, which leaves the Remainder’R’ =  39.

3. Subtract the Remainder: 39 from the greatest 3 digit number:999 ,which gives the required Greatest 3 digit number that is divisible exactly by the given numbers.  

Required Greatest 3 digit number = Greatest 3 digit number – Remainder.

  =  999 – 39 = 960 ,which is divisible exactly by the given numbers.