Question

If \[L.C.M.\left\{ {1,2,3, \ldots ,200} \right\} = b \times L.C.M.\left\{ {102,103,...,200} \right\}\], then \[b = \]

Answer 1

Hint: We will assume that \[LCM\left\{ {102,103,..,200} \right\}\] to be \[x\] and \[LCM\left\{ {1,2,3,...,99,100,101,102,...,200} \right\}\] to be \[y\]. We will simplify the equation and express \[b\] in terms of \[x\] and \[y\]. We will find the value of \[x\]and \[y\] using logic and the formula for the Least Common Multiple.

Formulas used:We will use the formula \[LCM\left\{ {a,b,c} \right\} = LCM\left\{ {c,LCM\left\{ {a,b} \right\}} \right\}\].
Complete step-by-step answer:
The L.C.M. of 2 or more numbers is the least common multiple of those numbers.
We have the equation
\[ \Rightarrow L.C.M.\left\{ {1,2,3, \ldots ,200} \right\} = b \times L.C.M.\left\{ {102,103,...,200} \right\}\]
On dividing both sides by \[L.C.M.\left\{ {102,103,...,200} \right\}\], we get
\[ \Rightarrow \dfrac{{L.C.M.\left\{ {1,2,3, \ldots ,200} \right\}}}{{L.C.M.\left\{ {102,103,...,200} \right\}}} = \dfrac{{b \times L.C.M.\left\{ {102,103,...,200} \right\}}}{{L.C.M.\left\{ {102,103,...,200} \right\}}}\]
\[ \Rightarrow \dfrac{{L.C.M.\left\{ {1,2,3, \ldots ,99,100,101,102,...,200} \right\}}}{{L.C.M.\left\{ {102,103,...,200} \right\}}} = b\left( * \right)\]
We will assume that \[LCM\left\{ {102,103,..,200} \right\} = x\] and \[LCM\left\{ {1,2,3,...,99,100,101,102,...,200} \right\} = y\] .
We know that some or other multiple of all the numbers lying between 1 and 100 will lie between 101 and 200. For example, the \[{51^{th}}\] multiple of 2 is 102 and it lies between 101 and 200, the 4th multiple of 30 is 120 and it lies between 101 and 200, the 2nd multiple of 99 is 198 and lies between 192 and 200…and so on.
So, we can safely say that
\[ \Rightarrow L.C.M.{\rm{ }}\left\{ {1,{\rm{ }}2,{\rm{ }}3, \ldots ,{\rm{ 200}}} \right\} = L.C.M.\left\{ {101,102,103,...,200} \right\}{\rm{ }}\left( 1 \right)\]
We know that \[LCM\left\{ {a,b,c} \right\} = LCM\left\{ {c,LCM\left\{ {a,b} \right\}} \right\}\], so \[ \Rightarrow L.C.M.\left\{ {101,102,103,...,200} \right\} = LCM\left\{ {101,LCM\left\{ {102,103,..,200} \right\}} \right\}\]
\[ \Rightarrow L.C.M.\left\{ {101,102,103,...,200} \right\} = LCM\left\{ {101,x} \right\}{\rm{ }}\left( 2 \right)\]
101 is a prime number and no multiple of 101 lies between 102 and 200. So,
\[ \Rightarrow L.C.M.\left\{ {101,y} \right\} = 101 \times x{\rm{ }}\left( 3 \right)\]
We will substitute equation (1) in equation (2):
\[ \Rightarrow L.C.M.\left\{ {101,102,103,...,200} \right\} = 101x{\rm{ }}\left( 4 \right)\]
We will substitute equation (4) in equation (1)
\[ \Rightarrow L.C.M.{\rm{ }}\left\{ {1,{\rm{ }}2,{\rm{ }}3, \ldots ,{\rm{ 200}}} \right\} = 101x\]
We will substitute \[101x\] for \[L.C.M.{\rm{ }}\left\{ {1,{\rm{ }}2,{\rm{ }}3, \ldots ,{\rm{ 200}}} \right\}\] and \[x\] for \[LCM\left\{ {102,103,..,200} \right\}\] in equation (*):
\[ \Rightarrow \dfrac{{101x}}{x} = b\]
\[ \Rightarrow 101 = b\]
\[\therefore\] The value of \[b\] is 101.

Note: The least common multiple of 2 numbers is the absolute value of their product divided by their greatest common divisor:
\[LCM\left( {ab} \right) = \dfrac{{\left| {ab} \right|}}{{\gcd \left( {a,b} \right)}}\]. The Least Common Multiple of a prime number (say \[p\]) with another number (say \[q\] )that is not its multiple is the product of the 2 numbers (\[pq\])