How to solve the following two questions Q1. Find the greatest number that will divide 640, 710 and 1526 so as to leave the reminders 11, 7 and 9 respectively? Q2. 10% of a mixture of alcohol and water replaced with water. If the resulting ratio of alcohol and water in the new mixture is 2 : 5 , what was the alcohol and water in the ratio in the original solution? Both are from Career launcher take off. The solutions given are very confusing |
|||
|
|
 |
when we divide 640, 710 and 1526 by a given number it leave the reminders 11, 7 and 9 respectively.....
so the given number will exactly divide (640-11) , (710-7) , (1526-9)
ie) it will exactly divide 629 , 703 , 1517
nd thaat no will be the hcf of these 3 number ....
which comes out be 37
correct answer is 37 for the first qus
Dont know
but how did you get
I would like to know the process rather than the answer
i think incognto's methid is good enough
Property 1:
The greatest number that divides each of a, b and c leaving the same remainder in each case is H.C.F. of |a – b|, |b – c|, |c – a|.
Example 1: The greatest number that will divide 132, 100 and 36 leaving the same remainder in each case is:
a) 8 b) 24 c) 32 d) 64
Solution:
In the given example, a = 132, b = 100 and c = 36
|a – b| = |132 – 100| = 32
|b – c| = |100 – 36| = 64
|c – a| = |36 – 132| = 96
Hence the required number = H.C.F. of 32, 64 and 96 which is equal to 32.
Hence 32 is the greatest number that divides each of the numbers 132, 100 and 36 to
leave the same remainder in each case and it can be verified that the remainder is 4 in
each case.
Example 2: Find the largest number that will divide 171, 333 and 576 to leave exactly
the same remainder in each case.
a) 9 b) 162 c) 27 d) 81
Solution:
In the given example, a = 171, b = 333 and c = 576
|a – b| = |171 – 333| = 162
|b – c| = |333 – 576| = 243
|c – a| = |576 – 171| = 405
Hence the required number = H.C.F. of 162, 243 and 405 which is equal to 81.
Hence 81 is the greatest number that divides each of the numbers 171, 333 and 576 to
leave the same remainder in each case and it can be verified that the remainder is 9 in
each case.
Thank You
Ravi Raja
n/a
Property 2:
The greatest number that divides each of a, b and c to leave remainders p, q and r respectively is H.C.F. of (a – p), (b – q), (c – r).
Example 1: Find the greatest number that will divide 887 and 1061 leaving remainders 7 and 5 respectively.
a) 32 b) 16 c) 176 d) 18
Solution:
In the given example, a = 887, b = 1061, p = 7 and q = 5
a – p = 887 – 7 = 880
b – q = 1061 – 5 = 1056
Hence the required number = H.C.F. of 880 and 1056 which is equal to 176.
Hence 176 is the greatest number that divides each of the numbers 887 and 1061 leaving remainders 7 and 5 respectively.
Example 2: Find the greatest number that will divide 114, 148 and 182 leaving remainders 2, 4 and 6 respectively.
a) 32 b) 16 c) 8 d) 28
Solution:
In the given example, a = 887, b = 1061, p = 7 and q = 5
a – p = 114 – 2 = 112
b – q = 148 – 4 = 144
c – r = 182 – 6 = 176
Hence the required number = H.C.F. of 112, 144 and 176 which is equal to 16.
Hence 16 is the greatest number that divides each of the numbers 114, 148 and 182 leaving remainders 2, 4 and 6 respectively.
Thank You
Ravi Raja
n/a
Property 3:
The least number which when divided by a, b, c leaves the same remainder ‘r’ in each case is {L.C.M. of a, b and c} + r.
Example 1: Find the least number which when divided by 16, 18 and 20 leaves a remainder 1 in each case.
a) 359 b) 361 c) 719 d) 721
Solution:
In the given example, a = 16, b = 18, c = 20 and r = 1
L.C.M. of 16, 18 and 20 = 720
Hence the required number is 720 + 1 = 721.
Example 2: Find the least number which when divided by 2, 3, 4, 5, 6, 7, 8, 9 and 10 leaves a remainder 1 in each case.
a) 2159 b) 2561 c) 2519 d) 2521
Solution:
The given property holds for any set of values of a, b and c.
Hence in this example, the required number is
{L.C.M. of 2, 3, 4, 5, 6, 7, 8, 9 and 10} + 1 = 2520 + 1 = 2521
Thank You
Ravi Raja
n/a
Property 4:
The least number which when divided by each of a, b and c leaves remainders p, q and r respectively, such that a – p = b – q = c – r = k (say), is {L.C.M. of a, b and c} – k.
(Note that in these type of problems, the remainders are such that the difference between the divisors and the remainders is the same in each case. This will become easier to understand once you go through the following two examples)
Example 1: Find the least number which when divided by 7, 8, 9 and 10 leaves 5, 6, 7, and 8 as remainders respectively.
a) 2518 b) 2522 c) 2528 d) None of these
Solution:
In the given example, a = 7, b = 8, c = 9, d = 10, p = 5, q = 6, r = 7, s = 8 and k = 2
Note that the difference between the divisors a, b, c, d and their respective remainders p, q, r and s is the same in each case and this difference = 2.
Hence the required number is {L.C.M. of 7, 8, 9 and 10} – 2 = 2520 – 2 = 2518
Example 2: 20, 25, 30, 36 and 48 divides a number. The respective remainders obtained are found to be 15, 20, 25, 31 and 43. Find the least such number
a) 3600 b) 3605 c) 3595 d) None of these
Solution:
Similar to the above question, note that the difference between the divisors and their respective remainders is the same in each case and this difference = 5.
Hence the required number is {L.C.M. of 20, 25, 30, 36 and 48} – 5 = 3600 – 5 = 3595
Thank You
Ravi Raja
n/a
Thanks Ravi for explaining everything in such a great detail with the properties
Hi
Ravisir thanks again for the solutions and answers
I was checking your profile and you have mentioned your Occupation as
Teacher but unfortunately you have not mentioned the location. If you are providing private tuition then please let the forum members know where you are located so that any body in that location can contact you
Post new comment