Aptitude - Numbers - Discussion
Discussion Forum : Numbers - General Questions (Q.No. 46)
46.
In dividing a number by 585, a student employed the method of short division. He divided the number successively by 5, 9 and 13 (factors 585) and got the remainders 4, 8, 12 respectively. If he had divided the number by 585, the remainder would have been
Answer: Option
Explanation:
5 | x z = 13 x 1 + 12 = 25
--------------
9 | y - 4 y = 9 x z + 8 = 9 x 25 + 8 = 233
--------------
13| z - 8 x = 5 x y + 4 = 5 x 233 + 4 = 1169
--------------
| 1 -12
585) 1169 (1
585
---
584
---
Therefore, on dividing the number by 585, remainder = 584.
Discussion:
39 comments Page 3 of 4.
Anujan said:
9 years ago
Thank you very much @Manideep.
(1)
SMcxlvii said:
9 years ago
Hope, my solution will be logical:
Suppose, the number is G.
According to the question:
G = 5a + 4 [where, a is an integer].
G = 9b + 8 [where, b is an integer].
G = 13c + 12 [where, c is an integer].
Now,
5a + 4 = G.
=> 5a + 5 - 1 = G.
=> 5(a + 1) = G + 1.
9b + 8 = G.
=> 9(b+) = G + 1.
13c + 12 = G.
=> 13(c + 1) = G+1.
Here, [5(a + 1)], [9(b + 1)], and [13(c + 1)] have same value [i.e: (G+1)].
So, the LCM of [5(a + 1)], [9(b + 1)], and [13(c + 1)] will also be (G + 1).
In other words,
LCM of [5(a + 1)], [9(b + 1)], and [13(c + 1)] = 585(a + 1)(b + 1)(c + 1).
So, 585(a + 1)(b + 1)(c + 1) = (G + 1).
Or, G = 585(a + 1)(b + 1)(c + 1) - 1.
Now, The number divided by 585.
=> G/585 = [585(a + 1)(b + 1)(c + 1) - 1]/585.
= [585(a + 1)(b + 1)(c + 1) - 585 + 584]/585.
= [{585(a + 1)(b + 1)(c + 1) - 1} + 584] = {(a + 1)(b + 1)(c + 1) - 1} + 584/585.
Surely {(a + 1)(b + 1)(c + 1) - 1} is an integer. So, the remainder is 584.
Then, the answer is 584.
Suppose, the number is G.
According to the question:
G = 5a + 4 [where, a is an integer].
G = 9b + 8 [where, b is an integer].
G = 13c + 12 [where, c is an integer].
Now,
5a + 4 = G.
=> 5a + 5 - 1 = G.
=> 5(a + 1) = G + 1.
9b + 8 = G.
=> 9(b+) = G + 1.
13c + 12 = G.
=> 13(c + 1) = G+1.
Here, [5(a + 1)], [9(b + 1)], and [13(c + 1)] have same value [i.e: (G+1)].
So, the LCM of [5(a + 1)], [9(b + 1)], and [13(c + 1)] will also be (G + 1).
In other words,
LCM of [5(a + 1)], [9(b + 1)], and [13(c + 1)] = 585(a + 1)(b + 1)(c + 1).
So, 585(a + 1)(b + 1)(c + 1) = (G + 1).
Or, G = 585(a + 1)(b + 1)(c + 1) - 1.
Now, The number divided by 585.
=> G/585 = [585(a + 1)(b + 1)(c + 1) - 1]/585.
= [585(a + 1)(b + 1)(c + 1) - 585 + 584]/585.
= [{585(a + 1)(b + 1)(c + 1) - 1} + 584] = {(a + 1)(b + 1)(c + 1) - 1} + 584/585.
Surely {(a + 1)(b + 1)(c + 1) - 1} is an integer. So, the remainder is 584.
Then, the answer is 584.
(3)
Kalkesh said:
1 decade ago
Please any one explain me in easy way.
(1)
Siddharth said:
1 decade ago
@Saurabh.
It looked very simple. Thanks for making it look easy but can you explain what is the logic we have to keep in mind behind this.
It looked very simple. Thanks for making it look easy but can you explain what is the logic we have to keep in mind behind this.
Saurabh said:
1 decade ago
5-4=1
9-8=1
13-12=1
585-?=1
So the number is 584.
9-8=1
13-12=1
585-?=1
So the number is 584.
(5)
Manjeet SIngh said:
1 decade ago
@Maindeep.. How come assumed x=5a+4,a=9b+8,b=13c+12.
Because it should have been like this x=5a+4=9b+8=13c+12=585d+r.
Where r is the required remainder ...So there will be 3 equations to solve for a, b, c and x but unknown are for.(I am only confused about your assumption..please through some light on that).
Because it should have been like this x=5a+4=9b+8=13c+12=585d+r.
Where r is the required remainder ...So there will be 3 equations to solve for a, b, c and x but unknown are for.(I am only confused about your assumption..please through some light on that).
Rahul said:
1 decade ago
@Sudhir
The question was to find the remainder when you divide the number by 585 but you used the method to calculate the number [required number =N*lcm(5,8,13)-1 ].
Can you explain what have you done ?
The question was to find the remainder when you divide the number by 585 but you used the method to calculate the number [required number =N*lcm(5,8,13)-1 ].
Can you explain what have you done ?
Ragul said:
1 decade ago
Divide 5 you get reminder as 4 and 9 you get 8 and 13 you get 12.
585+584 = 1169/5 = 4.
1+1+6+9 = 17/9 = 8.
Simply you get.
585+584 = 1169/5 = 4.
1+1+6+9 = 17/9 = 8.
Simply you get.
Rahul said:
1 decade ago
@Sudhir : How come we determine the value of N?
Surya said:
1 decade ago
How can we take the value of the last quotient as 1?
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