Finding Greatest number which is exactly divided and When divided leaves remainders.

Finding Greatest number using H.C.F in below cases:

(i) Finding Greatest number which is divisible exactly or leaves no remainder.

(ii) Finding Greatest number when on divided by numbers a, b, c  leaves Remainder’K’ in each case.

(iii) Finding Greatest number when on divided leaves Remainders x, y, z respectively.

 

 

Case(i) Steps to find greatest number, when on divided leaves no remainder:

We need to find the H.C.F of the given numbers: a, b, c which gives the greatest number when on divided by the numbers leaves no remainder.

Required Greatest number = H.C.F( a, b, c ).

 

Find the greatest number which is divisible exactly:

1) 1365, 1560, 1755.

Solution:

Given: Numbers = 1365, 1560, 1755.

We need to find the H.C.F of the numbers: H.C.F( 1365, 1560, 1755 ) = 195.

Greatest number = H.C.F( 1365, 1560, 1755 ) = 195.

 

2) 96, 528, 792

Solution:

Given: Numbers = 96, 528, 792.

We need to find the H.C.F(96, 528, 792 ) = 24.

Greatest number = H.C.F(96, 528, 792 ) = 24.

 

 

Case(ii) Steps to find Greatest number, when on divided by the numbers    a, b, c  leaves Remainder x, y, z respectively.

1. Find the Difference between the numbers and the remainders: (a – x) ,    (b – y) , (c – z ).

2. Find the H.C.F of the difference of the numbers and the remainders, which gives the required Greatest number when on divided leaves the remainders x, y, z respectively.

Greatest number = H.C.F ( a-x , b-y, c-z ).

 

1) Find the greatest number that will divide 728 ,900 leaving the remainders 8 ,4 respectively?

Solution:

Given: Numbers: a, b = 728, 900; Remainders: x, y = 8, 4.

1. Difference between the numbers and the remainders: a – x, b – y .

a – x = 728 – 8 = 720 ;  b – y = 900 – 4 = 896.

2. Find out H.C.F( 720, 896 ) = 16.

Greatest number = H.C.F( 720, 896 ) = 16.

 

2) Find the greatest number when on divided by 38, 45, 52 leaves remainders 2, 3, 4 respectively?

Solution:

Given: Numbers: a, b, c = 38, 45, 52; Remainders: x, y, z = 2, 3, 4.  

1. Difference between the numbers and the remainders: a – x , b – y , c – z ;

a – x = 38 – 2 = 36 ; b – y = 45 – 3 = 42 ; c – z = 52 – 4 = 48.

2. Find out the H.C.F( 36, 42, 48 ) = 6.

Greatest number = H.C.F( 36, 42, 48 ) = 6.

 

 

Case(iii) Steps to find the Greatest number, when on divided by the numbers a, b, c leaves same remainder ‘ K’ in each case.

1. Find the difference between the numbers: a, b, c and the remainder ’K’ :

a – K , b – k , c – k.

2. Find out the H.C.F of the difference of the numbers: a, b, c and the remainder ‘K’

i.e; H.C.F( a – K , b – K , c – K ).

Greatest Number = H.C.F( a – K , b – K , c – K ), when on divided by the numbers: a, b, c leaves the remainder ‘K’ in each case.

 

Find the greatest number when on divided by the numbers 260, 720, 1410 so as to leave the remainder 7 in each case?

Solution:

Given: Numbers: a, b, c = 260, 720, 1410; Remainder ‘K’ = 7.

1. Difference between the numbers: a – K , b – K , c – k

    a – K = 260 – 7 = 253 ; b – K = 720 – 7 = 713 ; c – K = 1410 – 7 = 1403.

2. Finding H.C.F( a-k , b-k, c-k ) = H.C.F ( 253, 713, 1403 ) = 23.

Factors of 253  = 11 * 23,

Factors of 713  = 31 * 23,

Factors of 1403 = 61 * 23.   

H.C.F ( 253, 713, 1403 ) = 23.

Greatest number = H.C.F ( 253, 713, 1403 ) = 23.