Problem - Find the greatest number, which will divide 215,167 and 135 so as to leave the same remainder in each case.
I understood that we have to take the GCD of each number. But 167 is a prime. How do we solve this?
Number properties
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134 will leave the same remainder when divided by 215,167 and 135
from 135 to 215, remainders will be different
ans 134
from 135 to 215, remainders will be different
ans 134
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let k be the number which will leave same remainder 'r' in each case when divided 215,167 and 135 resp.Rajasamantray wrote:Problem - Find the greatest number, which will divide 215,167 and 135 so as to leave the same remainder in each case.
I understood that we have to take the GCD of each number. But 167 is a prime. How do we solve this?
now, 215=k*q1+r;----1)
167=k*q2+r;----------2)
135=k*q3+r;----------3)
subtracting 2 from 1 we have;
48=k*(q1-q2);-------4)
subtracting 3 from 2 we have;
32=k*(q2-q3);--------5)
subtracting 3 from 1 we have;
80=k*(q3-q1);--------6)
k will be the hcf of 4,5,6 which is 16, hence the no. is 16...!!!
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134 leaves 81, 33 and 1 as remainder. That is not the answer. The remainder has to be the same.cans wrote:134 will leave the same remainder when divided by 215,167 and 135
from 135 to 215, remainders will be different
ans 134
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Answer is 16
Rule- Greatest number with which if we divide p,q,r and it leaves same remainder in each case...
Number is of form = HCF of (p-q),(p-r)
Therefore, HCF of (215-167), (213-135) = HCF of 48, 80 = 16.
215 / 16 ---> Remainder is 7
167 / 16 ---> Remainder is 7
135 / 16 ---> Remainder is 7
Rule- Greatest number with which if we divide p,q,r and it leaves same remainder in each case...
Number is of form = HCF of (p-q),(p-r)
Therefore, HCF of (215-167), (213-135) = HCF of 48, 80 = 16.
215 / 16 ---> Remainder is 7
167 / 16 ---> Remainder is 7
135 / 16 ---> Remainder is 7
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Nice Explanation manpsigh!!!!
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