What is the greatest number that will divide 172, 205 and 304 so as to leave the same remainder in each case?

Question

If 'P' is the greatest number, that divides 9086, 12326 and 23126 leaving the same remainder in each case.
What is the sum of digits in P?

Answer 1

The answer is 33

Lets put them into equations
172 = D*q1 + r1;
205 = D*q2 + r2;
304 = D*q3 + r3;

and r1=r2=r3=r

so.
D*(q2-q1)= 33
D*(q3-q2)= 99
D*(q3-q1)= 132

For the greatest value of D take the HCF of 33, 99 and 132 which is 33.

Answer 2

shortcut

step1 let assume a=172
b=205
c=304
step2 take difference
(b-a)=33
(c-b)=99
(c-a)=132
now take a HCF of these numbers(33,99,132)
HCF=33(ans)

Answer 3

33 is the answer
Explanation:
The greatest number that will divide 172,205 and 304 leaving the same remainder in each
case is HCF ((205-172), (304-205), (304-172))
= HCF of 33, 99, and 132
33=31x11
99=3 x 3 x 11
132=3 x 2 x 2 x11
Required number =3 x 11=33