To determine the lowest possible value of [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] is a common multiple of 4 and 9, and [tex]\(q\)[/tex] is a common factor of 72 and 120, we will work through it step-by-step.
1. Find the Least Common Multiple (LCM) of 4 and 9:
The LCM of two numbers is the smallest positive integer that is divisible by both numbers.
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(9 = 3^2\)[/tex]
To find the LCM, we take the highest powers of all prime factors involved:
[tex]\[
\text{LCM}(4, 9) = 2^2 \times 3^2 = 4 \times 9 = 36
\][/tex]
Therefore, [tex]\(p = 36\)[/tex].
2. Find the Greatest Common Divisor (GCD) of 72 and 120:
The GCD of two numbers is the largest positive integer that divides both numbers exactly.
- Prime factorizations:
- [tex]\(72 = 2^3 \times 3^2\)[/tex]
- [tex]\(120 = 2^3 \times 3 \times 5\)[/tex]
The GCD is obtained by taking the lowest powers of all prime factors that appear in both factorizations:
[tex]\[
\text{GCD}(72, 120) = 2^3 \times 3^1 = 8 \times 3 = 24
\][/tex]
Therefore, [tex]\(q = 24\)[/tex].
3. Calculate the lowest possible value of [tex]\(\frac{p}{q}\)[/tex]:
Now that we have [tex]\(p = 36\)[/tex] and [tex]\(q = 24\)[/tex], we can find
[tex]\[
\frac{p}{q} = \frac{36}{24} = 1.5
\][/tex]
Thus, the lowest possible value of [tex]\(\frac{p}{q}\)[/tex] is [tex]\(\boxed{1.5}\)[/tex].
