To find the value of [tex]\( x \)[/tex] that is a common multiple of 6 and 7, and also a common factor of 252 and 420, while satisfying [tex]\( 50 < x < 150 \)[/tex], follow these steps:
1. Find the Least Common Multiple (LCM) of 6 and 7:
- The LCM of two numbers is the smallest multiple that is divisible by both of them.
- Since 6 and 7 are coprime (which means their greatest common divisor (GCD) is 1), their LCM is simply their product:
[tex]\[
\text{LCM}(6, 7) = 6 \times 7 = 42
\][/tex]
2. Identify the multiples of 42 within the range [tex]\( 50 < x < 150 \)[/tex]:
- The multiples of 42 are:
[tex]\[
42, 84, 126, 168, \ldots
\][/tex]
- We need to exclude 42 because it is not greater than 50, and 168 because it is not less than 150. This leaves us with:
[tex]\[
84 \quad \text{and} \quad 126
\][/tex]
3. Find the common factors of 252 and 420:
- The common factors of two numbers are the numbers that divide both of them without leaving a remainder.
- The factors of 252 are: 1, 2, 3, 4, 6, 7, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252
- The factors of 420 are: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420
- The common factors of 252 and 420 are:
[tex]\[
1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
\][/tex]
4. Determine the value of [tex]\( x \)[/tex] that meets all the criteria:
- We need [tex]\( x \)[/tex] to be one of the multiples of 42 in the range 50 to 150, which are:
[tex]\[
84 \quad \text{and} \quad 126
\][/tex]
- Out of these, we need [tex]\( x \)[/tex] to also be a common factor of 252 and 420. From our list of common factors, 84 is a common factor whereas 126 is not.
Therefore, the only value for [tex]\( x \)[/tex] that satisfies all conditions is:
[tex]\[
x = 84
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{84} \)[/tex].
