The smallest number which when divided by 20, 25, 35 and 40 leaves a remainder of 14, 19, 29 and 34 respectively, is 1394. How?

By BYJU'S Exam Prep

Updated on: September 25th, 2023

1394 is the smallest number divisible by 10, 25, 35 and 40 leaving a remainder of 14, 19, 29 and 34 respectively. To prove this statement, we will subtract the reminders from these numbers and then find the LCM of the numbers derived. Check the detailed solution below.

How does 1394 when divided by 20, 25, 35 and 40 leaves a remainder of 14, 19, 29 and 34?

To prove that 1394 is the least number that will leave remainders 14, 19, 29, and 34 when divided by 20, 25, 35, and 40 respectively, we will first find a common value by subtracting remainders from their respective numbers.

20–14 = 6
25–19 = 6
35–29 = 6
40–34 = 6

Hence, the common value is 6. (A)

Now, that we have a common value, we will find the LCM of 20, 25, 35 and 40 –

20 = 2^2 x 5

25 = 5^2

35 = 5×7

40 = 2^3 x 5

LCM = 2^3 x 5^2 x 7 = 1400 (B)

Now, again, we will subtract the number derived in (A) from (B) –

1400 – 6 = 1394

Thus, it is proved that the smallest number divisible by 10, 25, 35 and 40 leaves a remainder of 14, 19, 29 and 34 respectively is 1394 by calculating the LCM and subtracting the common value from it.

Summary:

The smallest number which when divided by 20, 25, 35 and 40 leaves a remainder of 14, 19, 29 and 34 respectively, is 1394. How?

By calculating the LCM and subtracting the common value from it, it was proved that the smallest number divisible by 10, 25, 35, and 40 leaves a remainder of 14, 19, 29, and 34 respectively is 1394. In such questions, candidates should first find the common value by subtracting numbers from the remainder and then finding the LCM. Subtracting LCM with the common value will give you your answer.

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