The HCF and LCM of two number are 2x3 and 2 squaredx3 power of 3x 7 squared. Given that one of the numbers is 252, find the other possible number.
To find the other number, we can use the relationship between the highest common factor (HCF) and the least common multiple (LCM) of two numbers, along with the given number. The product of two numbers is equal to the product of their HCF and LCM.
Let's denote the numbers as A and B, where A is 252 and B is the number we want to find. Their HCF is \(2 \times 3\), and their LCM is \(2^2 \times 3^3 \times 7^2\).
Using the formula:
\[ \text{HCF(A, B)} \times \text{LCM(A, B)} = A \times B \]
Plug in the values:
\[ (2 \times 3) \times (2^2 \times 3^3 \times 7^2) = 252 \times B \]
Now simplify and calculate the values:
\[ 6 \times (4 \times 27 \times 49) = 252 \times B \]
\[ 6 \times 4 \times 27 \times 49 = 252 \times B \]
\[ 24 \times 27 \times 49 = 252 \times B \]
\[ (6 \times 4) \times (3^3) \times (7^2) = 252 \times B \]
\[ (6 \times 4) \times (3 \times 9) \times (49) = 252 \times B \]
\[ 24 \times 27 \times 49 = 252 \times B \]
\[ 24 \times (3^3) \times 7^2 = 252 \times B \]
\[ (8 \times 3) \times (3 \times (3 \times 3)) \times (7 \times 7) = 252 \times B \]
\[ 24 \times 27 \times 49 = 252 \times B \]
First, factorize 252 into its prime factors to help with later calculations:
\[ 252 = 2^2 \times 3^2 \times 7 \]
Now, calculate the value of \(24 \times 27 \times 49\):
\[ 24 \times 27 \times 49 = 2^3 \times 3^1 \times 3^3 \times 7^2 \]
\[ = 2^3 \times 3^4 \times 7^2 = 8 \times 81 \times 49 \]
\[ = 8 \times 81 \times 49 \]
\[ = 648 \times 49 \]
\[ = 31752 \]
So we have:
\[ 31752 = 252 \times B \]
Now divide both sides by 252 to find B:
\[ B = 31752 / 252 \]
\[ B = 2^3 \times 3^4 \times 7^2 รท 2^2 \times 3^2 \times 7 \]
\[ B = 2^{3-2} \times 3^{4-2} \times 7^{2-1} \]
\[ B = 2^1 \times 3^2 \times 7^1 \]
\[ B = 2 \times 9 \times 7 \]
\[ B = 18 \times 7 \]
\[ B = 126 \]
So the other possible number is 126.