Question

The smallest number which when divided by 20, 25, 35 and 40 and leaves a remainder of 14, 19, 29 and 34 respectively is
A) 1394
B) 1404
C) 1664
D) 1406

Answer 1

Hint:
Here, we will first find the least common multiple, which is the smallest positive number that is a multiple of two or more numbers. We will find the remainder just by subtracting the given numbers from the divisors. Then we will subtract the remainder left by dividing the given divisor from the obtained L.C.M. to find the required value.

Complete step by step solution:
Given that the numbers are 20, 25, 35 and 40.
We know that the least common multiple L.C.M. is the smallest positive number that is a multiple of two or more numbers.
We will now find the least common multiple of the given numbers, 20, 25, 35 and 40.

We will find the product of the above multiplies to find the least common multiple.
\[
  {\text{L.C.M}} = 2 \times 2 \times 2 \times 5 \times 5 \times 7 \\
   = 1400 \\
 \]
Since each of the given numbers are greater than the divisors by small difference, so we can find the remainder just by subtracting the given numbers from the divisors.
Subtracting the number 14 from 20 to find the remainder, we get
\[20 - 14 = 6\]
Subtracting the number 19 from 25 to find the remainder, we get
\[25 - 19 = 6\]
Subtracting the number 29 from 35 to find the remainder, we get
\[35 - 29 = 6\]
Subtracting the number 34 from 40 to find the remainder, we get
\[40 - 34 = 6\]
Since we get 6 from all of the above differences, we will now find the required number.
Subtracting the number 6 from the obtained least common multiple, we get
\[
   \Rightarrow 1400 - 6 \\
   \Rightarrow 1394 \\
 \]
Therefore, the smallest number which when divided by 20, 25, 35 and 40 and leaves a remainder of 14, 19, 29 and 34 respectively is 1394.

Hence, option A is correct.

Note:
Whenever such type of question comers try to find the common factor by calculating the L.C.M. after that we will calculate the remainder to find the smallest number which when divided by given number leaves the remainders 14, 19, 29 and34 respectively.