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Which greatest number of 4 digits is exactly divisible by 12,15,20 and 35.
A.9999
B.9660
C.9832
D.9860
Answer
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Hint: First we have to take the L.C.M of the given numbers in the question. L.C.M is the least common Multiple. It is the smallest positive number that is a multiple of two or more numbers. We know that the greatest 4 digits number is 9999 , after taking the L.C.M of the given numbers we have to divide it with 9999, and finally on subtracting the remainder from 9999, we will get the correct answer.
Complete step-by-step answer:
First we will take the L.C.M of the given numbers.
Given numbers are: $12,15,20\operatorname{and} 35$
Now we will take the L.C.M of the given numbers:
$
12 = 2 \times 2 \times 3 \\
15 = 5 \times 3 \\
20 = 2 \times 2 \times 5 \\
35 = 7 \times 5 \\
$
Now L.C.M =$4 \times 3 \times 5 \times 7 = 420$ \[\]
Least common Multiple of the numbers is = 420
We know that the greatest 4 digits number is 9999
Now from the given solution hint:
We have divided the number 9999 by 420
Now we will get the remainder = 339
So largest number of four digits is exactly divisible by 12,15,20 and 35will be:
$
\Rightarrow 9999 - 339 \\
\Rightarrow 9660 \\
$
So the largest number of four digits is exactly divisible by 12,15,20 and 35 will be 9660.
Hence the correct answer is option B.
Note: We have to remember the concept of L.C.M which is least common Multiple. We know that the greatest 4 digits number is 9999, so by using the concept of L.C.M we will find the least common Multiple in the given numbers 12,15,20 and 35. After that by using the solution hint, we will proceed for the further steps. Thus we will get the correct answer to the given question.
Complete step-by-step answer:
First we will take the L.C.M of the given numbers.
Given numbers are: $12,15,20\operatorname{and} 35$
Now we will take the L.C.M of the given numbers:
$
12 = 2 \times 2 \times 3 \\
15 = 5 \times 3 \\
20 = 2 \times 2 \times 5 \\
35 = 7 \times 5 \\
$
Now L.C.M =$4 \times 3 \times 5 \times 7 = 420$ \[\]
Least common Multiple of the numbers is = 420
We know that the greatest 4 digits number is 9999
Now from the given solution hint:
We have divided the number 9999 by 420
Now we will get the remainder = 339
So largest number of four digits is exactly divisible by 12,15,20 and 35will be:
$
\Rightarrow 9999 - 339 \\
\Rightarrow 9660 \\
$
So the largest number of four digits is exactly divisible by 12,15,20 and 35 will be 9660.
Hence the correct answer is option B.
Note: We have to remember the concept of L.C.M which is least common Multiple. We know that the greatest 4 digits number is 9999, so by using the concept of L.C.M we will find the least common Multiple in the given numbers 12,15,20 and 35. After that by using the solution hint, we will proceed for the further steps. Thus we will get the correct answer to the given question.
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