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The least multiple of 13 which when divided by 4, 5, 6, 7 leaves remainder 3 in each case is        (SSC CGL 2nd Sit. 2012)
  • a)
    3780
  • b)
    3783
  • c)
    2520
  • d)
    2522
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
The least multiple of 13 which when divided by 4, 5, 6, 7 leaves remai...
LCM of 4, 5, 6 and 7 = 420
∴ Required number
= 420k + 3 which is exactly divisible by 13.
= 32 × 13k + 4k + 3
Hence, 4k + 3 should be divisible by 13 for some value of k.
For k = 9, 4k + 3 = 39 which is divisible by 13.
∴ Required number = 420 × 9 + 3 = 3783
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Community Answer
The least multiple of 13 which when divided by 4, 5, 6, 7 leaves remai...
To find the least multiple of 13 that leaves a remainder of 3 when divided by 4, 5, 6, and 7, we can use the concept of the least common multiple (LCM).

Step 1: Find the LCM of 4, 5, 6, and 7.

The LCM of 4, 5, 6, and 7 can be found by listing their multiples and finding the smallest common multiple.

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ...

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, ...

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, ...

From the lists above, we can see that the smallest common multiple is 60. Therefore, the LCM of 4, 5, 6, and 7 is 60.

Step 2: Find the smallest multiple of 13 that leaves a remainder of 3 when divided by 60.

To find the smallest multiple of 13 that leaves a remainder of 3 when divided by 60, we can start by finding the first multiple of 13 that is greater than 60.

Multiples of 13: 13, 26, 39, 52, 65, 78, ...

From the list above, we can see that the first multiple of 13 greater than 60 is 65.

Now, we can subtract 60 from 65 to find the remainder: 65 - 60 = 5.

Since the remainder is not 3, we need to keep subtracting 60 from the multiple of 13 until we get a remainder of 3.

65 - 60 = 5 (remainder is not 3)
65 - 60 - 60 = 5 - 60 = -55 (remainder is not 3)
65 - 60 - 60 - 60 = 5 - 60 - 60 = -115 (remainder is not 3)
65 - 60 - 60 - 60 - 60 = 5 - 60 - 60 - 60 = -175 (remainder is not 3)
65 - 60 - 60 - 60 - 60 - 60 = 5 - 60 - 60 - 60 - 60 = -235 (remainder is not 3)
65 - 60 - 60 - 60 - 60 - 60 - 60 = 5 - 60 - 60 - 60 - 60 - 60 = -295 (remainder is not 3)

Finally, when we subtract 60 from 65 six times, we get a remainder of 3: 65 - 60 - 60 - 60 - 60 -
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