Question

Find the HCF and LCM of 625, 1125 and 2125, using Fundamental Theorem of Arithmetic method.
A) 95625
B) 95425
C) 98621
D) 92536

Answer 1

Hint:
Here, we will use the fundamental theorem of arithmetic to find the factors. Then by using the factors, we will find the HCF by multiplying the factor which is common to all. We will then multiply the factors that are least common in all to get the LCM. Highest Common Factor is defined as the number which exactly divides all the numbers. Least Common Multiple is defined as a number which is exactly divisible by all the numbers.

Complete step by step solution:
We are given the numbers 625, 1125 and 2125.
We will find the factors of 625, 1125 and 2125.
Now, we will find the factor of 625 using the fundamental theorem of arithmetic.
$\begin{array}{*{35}{l}}
   5 | 625 \\
   5 | 125 \\
   5 | 25 \\
   5 | 5 \\
   {} | 1 \\
\end{array}$
Thus, the factor of 625 is \[5 \times 5 \times 5 \times 5\].
Now, we will find the factor of 1125 using the fundamental theorem of arithmetic.
$\begin{array}{*{20}{l}}
  5| {1125} \\
\hline
  5| {225} \\
\hline
  5| {45} \\
\hline
  3| 9 \\
\hline
  3| 3 \\
\hline
  {}| 1
\end{array}$
Thus, the factor of 1125 is \[5 \times 5 \times 5 \times 3 \times 3\] .

Now, we will find the factor of 2125 using the fundamental theorem of arithmetic.
$\begin{array}{*{35}{l}}
   5 | 2125 \\
   5 | 425 \\
   5 | 85 \\
   17 | 17 \\
   \,\,{} | 1 \\
\end{array}$
Thus, the factor of 2125 is \[5 \times 5 \times 5 \times 17\].
Now, by using the factors, we will find the H.C.F and L.C.M of 625, 1125 and 2125.
$625={{5}^{4}}$
\[1125 = {5^3} \times {3^2}\]
$2125 = {5^3} \times {17^1}$
This can be also represented as
\[625 = {5^4} \times {3^0} \times {17^0}\]
\[1125 = {5^3} \times {3^2} \times {17^1}\]
\[2125={{5}^{3}}\times {{3}^{0}}\times {{17}^{1}}\]
Now, Highest Common Factors is the smallest power of the common factors.
The common factors of 625, 1125 and 2125 are
\[625 = {5^4}\]
$1125={{5}^{3}}$
\[2125 = {5^3}\]
Thus, HCF of (625, 1125 and 2125) $={{5}^{3}}$
Thus, HCF of (625, 1125 and 2125) $=125$
Now, Least Common Multiple is the highest power of the prime factors.
The prime factors of 625, 1125 and 2125 are
\[625 = {5^4} \times {3^0} \times {17^0}\]
$1125={{5}^{3}}\times {{3}^{2}}\times {{17}^{1}}$
$2125={{5}^{3}}\times {{3}^{0}}\times {{17}^{1}}$
Thus, LCM of 625, 1125 and 2125 $={{5}^{4}}\times {{3}^{2}}\times {{17}^{1}}$
Simplifying the expression, we get
\[ \Rightarrow \] LCM of (625, 1125 and 2125) $=95625$
Therefore, the HCF and LCM of 625, 1125 and 2125, using Fundamental Theorem of Arithmetic method is 125 and 98625.

Thus, option (A) is the correct answer.

Note:
We know that the fundamental theorem of Arithmetic states that every prime number except 1 is either a prime number or it can be expressed in the form of primes. It can be also stated as the natural numbers can be expressed in the form of prime numbers. The fundamental theorem of arithmetic is also called a unique prime factorization theorem. Common Factor is the factor that is common to all the numbers whereas the prime factors are the factors, which is the product of the powers of the prime numbers.