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What is the sum of the digits of the least number, which when divided by 12,16 and 54, leaves the same remainder 7 in each case, and is also completely divisible by 13? (SSC Sub. Ins. 2018 )
- a)36
- b)16
- c)9
- d)27
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
What is the sum of the digits of the least number, which when divided ...
Solution:
We need to find the least number which when divided by 12, 16 and 54 leaves the same remainder of 7 in each case and is also completely divisible by 13.
Let the required number be 'N'. Then we can write:
N = LCM(12, 16, 54)k + 7
where k is some integer.
Finding LCM of 12, 16 and 54, we get:
LCM(12, 16, 54) = 432
So, N = 432k + 7
Now, we need to find the least value of k for which N is divisible by 13.
When N is divided by 13, we get a remainder of:
(432k + 7) mod 13 = (5k + 7) mod 13
For N to be divisible by 13, (5k + 7) must be a multiple of 13.
The first few values of k for which (5k + 7) is a multiple of 13 are:
k = 3, 16, 29, 42, ...
The least value of k for which N is divisible by 13 is k = 3.
So, N = 432(3) + 7 = 1303
The sum of the digits of 1303 is:
1 + 3 + 0 + 3 = 7
Therefore, the required sum is 7 and the correct answer is option B.
We need to find the least number which when divided by 12, 16 and 54 leaves the same remainder of 7 in each case and is also completely divisible by 13.
Let the required number be 'N'. Then we can write:
N = LCM(12, 16, 54)k + 7
where k is some integer.
Finding LCM of 12, 16 and 54, we get:
LCM(12, 16, 54) = 432
So, N = 432k + 7
Now, we need to find the least value of k for which N is divisible by 13.
When N is divided by 13, we get a remainder of:
(432k + 7) mod 13 = (5k + 7) mod 13
For N to be divisible by 13, (5k + 7) must be a multiple of 13.
The first few values of k for which (5k + 7) is a multiple of 13 are:
k = 3, 16, 29, 42, ...
The least value of k for which N is divisible by 13 is k = 3.
So, N = 432(3) + 7 = 1303
The sum of the digits of 1303 is:
1 + 3 + 0 + 3 = 7
Therefore, the required sum is 7 and the correct answer is option B.
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Community Answer
What is the sum of the digits of the least number, which when divided ...
L.C.M. of 12, 16 and 54.
12 = 2 × 2 × 3.
16 = 2 × 2 × 2 × 2.
54 = 2 × 3 × 3 × 3
L.C.M. = 2 × 2 ×2 × 2 × 3 × 3 × 3 = 432
Remainder = 7.
So, required number = 432 + 7 = 439.
But this is not divisible by 13.
so, next number is 432 × 2 + 7 = 871.
Number 871 is divisible by 13.
Hence, required number is 871.
Sum of its digits = 8 + 7 + 1 = 16.
12 = 2 × 2 × 3.
16 = 2 × 2 × 2 × 2.
54 = 2 × 3 × 3 × 3
L.C.M. = 2 × 2 ×2 × 2 × 3 × 3 × 3 = 432
Remainder = 7.
So, required number = 432 + 7 = 439.
But this is not divisible by 13.
so, next number is 432 × 2 + 7 = 871.
Number 871 is divisible by 13.
Hence, required number is 871.
Sum of its digits = 8 + 7 + 1 = 16.
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