Question

Find the least 5 digit number which is exactly divisible by \[{\text{2}}0,{\text{25}},{\text{3}}0\]
A) \[10000\]
B) \[10200\]
C) \[30000\]
D) \[12000\]

Answer 1

Hint: First we know about the least five digit number. To get the least number, we find Least Common Multiple (LCM) of these numbers. Then multiply the LCM to the numbers that make it least five digit number. First we find the factors of given 3 numbers. Then we find LCM of three numbers.
In question LCM can be calculated by Prime Factorization. Steps for this method is
(i) Write all the prime factors of each number.
(ii) List all the prime numbers.
(iii) in last multiply the list of prime factors together

Complete step-by-step answer:
Given: numbers are \[{\text{2}}0,{\text{25}},{\text{3}}0\]
First we find factors of given numbers
\[{\text{2}}0 = {\text{ 1}} \times {\text{2}} \times {\text{2}} \times {\text{5}}\]
\[{\text{25}} = {\text{1}} \times {\text{5}} \times {\text{5}}\]
\[{\text{3}}0 = {\text{1}} \times {\text{2}} \times {\text{3}} \times {\text{5}}\]
\[ \Rightarrow {\text{LCM}} = {\text{ 1}} \times {\text{2}} \times {\text{2}} \times {\text{3}} \times {\text{5}} \times {\text{5}}\]
\[\therefore {\text{LCM}} = {\text{3}}00\]
Hence, LCM of \[{\text{2}}0,{\text{25}},{\text{3}}0\] is \[300\].
But the requirement of a minimum 5 digit number which can be divisible by \[{\text{2}}0,{\text{25}}\] and \[{\text{3}}0\].
We know that \[10000\] is atleast 5 digit number. It can be divisible by \[{\text{2}}0, {\text{25}}\] but can't be divisible by \[{\text{3}}0\]. For make divisible, we can find that higher quotient=\[34\]
Hence, the required least number which is exactly divisible by \[{\text{2}}0,{\text{25}},{\text{3}}0\]\[ = 34 \times 300 = 10200\]

Therefore, the correct option is B.

Note: The Least Common Multiple (LCM) is defined as the Lowest Common Multiple (LCM). It is also called Least Common Divisor (LCD). LCM for two integers a and b is given by LCM(a,b). The LCM is the smallest positive integer which is evenly divisible by both numbers or digits a and b. For example, LCM of \[2,3\] is denoted by LCM \[(2,3)\]. Value of \[LCM(2,3) = 6\]. Other example is LCM of is denoted by \[{\text{LCM}}\left( {{\text{6}},{\text{1}}0} \right)\] and value is \[{\text{3}}0\].
The LCM is calculated by following 5 different methods:
(1) Listing Multiples
(2) Prime Factorization
(3) Ladder Method
(4) Division Method
(5) Using the Greatest Common Factor GCF