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A certain number when divided by 247 leaves a remainder 17, another number when divided by 361 leaves a remainder 52. What is the remainder when the sum of these two numbers is divided by 19?
- a)8
- b)9
- c)12
- d)17
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A certain number when divided by 247 leaves a remainder 17, another n...
N1 = 247x + 17 and N2 = 361y + 52
View all questions of this test
N1 + N2 = (19 × 13 × x + 17) + (19 ×19 × y + 52)
Remainder when N1 + N2 is divided by 19,
=
=12
Most Upvoted Answer
A certain number when divided by 247 leaves a remainder 17, another n...
Given:
Number 1 divided by 247 leaves a remainder of 17.
Number 2 divided by 361 leaves a remainder of 52.
To find:
The remainder when the sum of these two numbers is divided by 19.
Solution:
Let's assume the two numbers to be x and y.
So, we have
x = 247a + 17 (a is some integer)
y = 361b + 52 (b is some integer)
Adding x and y, we get
x + y = 247a + 17 + 361b + 52
x + y = 247a + 361b + 69
Now, let's divide x + y by 19.
(247a + 361b + 69) / 19
= (247a / 19) + (361b / 19) + (69 / 19)
Since 247 and 19 are co-prime, (247a / 19) leaves the same remainder as (a / 19).
Similarly, (361b / 19) leaves the same remainder as (b / 19).
So, the remainder of (247a / 19) + (361b / 19) + (69 / 19) is the same as the remainder of (a / 19) + (b / 19) + (69 / 19).
Now, let's find the remainders of a/19 and b/19.
To find the remainder of a/19, we need to find a number which when multiplied by 19 and added to 17 gives us x.
Similarly, to find the remainder of b/19, we need to find a number which when multiplied by 19 and added to 52 gives us y.
We can solve these using the Chinese remainder theorem, which gives us:
a = 10 mod 19
b = 3 mod 19
Substituting these values in (a / 19) + (b / 19) + (69 / 19), we get:
(a / 19) + (b / 19) + (69 / 19)
= (10 / 19) + (3 / 19) + (69 / 19)
= 12 mod 19
Therefore, the remainder when the sum of the two numbers is divided by 19 is 12 (option C).
Number 1 divided by 247 leaves a remainder of 17.
Number 2 divided by 361 leaves a remainder of 52.
To find:
The remainder when the sum of these two numbers is divided by 19.
Solution:
Let's assume the two numbers to be x and y.
So, we have
x = 247a + 17 (a is some integer)
y = 361b + 52 (b is some integer)
Adding x and y, we get
x + y = 247a + 17 + 361b + 52
x + y = 247a + 361b + 69
Now, let's divide x + y by 19.
(247a + 361b + 69) / 19
= (247a / 19) + (361b / 19) + (69 / 19)
Since 247 and 19 are co-prime, (247a / 19) leaves the same remainder as (a / 19).
Similarly, (361b / 19) leaves the same remainder as (b / 19).
So, the remainder of (247a / 19) + (361b / 19) + (69 / 19) is the same as the remainder of (a / 19) + (b / 19) + (69 / 19).
Now, let's find the remainders of a/19 and b/19.
To find the remainder of a/19, we need to find a number which when multiplied by 19 and added to 17 gives us x.
Similarly, to find the remainder of b/19, we need to find a number which when multiplied by 19 and added to 52 gives us y.
We can solve these using the Chinese remainder theorem, which gives us:
a = 10 mod 19
b = 3 mod 19
Substituting these values in (a / 19) + (b / 19) + (69 / 19), we get:
(a / 19) + (b / 19) + (69 / 19)
= (10 / 19) + (3 / 19) + (69 / 19)
= 12 mod 19
Therefore, the remainder when the sum of the two numbers is divided by 19 is 12 (option C).
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A certain number when divided by 247 leaves a remainder 17, another number when divided by 361 leaves a remainder 52. What is the remainder when the sum of these two numbers is divided by 19?a) 8b) 9c) 12d) 17Correct answer is option 'C'. Can you explain this answer?
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