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Answer :
We start by finding the highest common factor (H.C.F.) for each set of numbers.
________________________________________
1. For the numbers [tex]\(120\)[/tex], [tex]\(150\)[/tex], and [tex]\(210\)[/tex]:
a) First, find the H.C.F. of [tex]\(120\)[/tex] and [tex]\(150\)[/tex]. It turns out that
[tex]\[
\text{H.C.F.}(120,150)=30.
\][/tex]
b) Next, find the H.C.F. of [tex]\(30\)[/tex] (the result above) and [tex]\(210\)[/tex]:
[tex]\[
\text{H.C.F.}(30,210)=30.
]
Thus, the overall H.C.F. is:
\[
\text{H.C.F.}(120,150,210)=30.
\][/tex]
According to the problem, this H.C.F. is expressed as:
[tex]\[
k^2-6=30.
\][/tex]
To find [tex]\(k\)[/tex], solve the equation:
[tex]\[
\begin{aligned}
k^2 - 6 &= 30,\\[1mm]
k^2 &= 30 + 6 = 36,\\[1mm]
k &= \sqrt{36}=6.
\end{aligned}
\][/tex]
So, the value of [tex]\(k\)[/tex] is [tex]\(6\)[/tex].
________________________________________
2. For the numbers [tex]\(17\)[/tex], [tex]\(23\)[/tex], and [tex]\(29\)[/tex]:
Since these are all prime numbers and do not have any common factors other than [tex]\(1\)[/tex]:
[tex]\[
\text{H.C.F.}(17,23,29)=1.
\][/tex]
________________________________________
Final Answers:
10. The value of [tex]\(k\)[/tex] is [tex]\(\boxed{6}\)[/tex].
11. The H.C.F. of [tex]\(17\)[/tex], [tex]\(23\)[/tex], and [tex]\(29\)[/tex] is [tex]\(\boxed{1}\)[/tex].
________________________________________
1. For the numbers [tex]\(120\)[/tex], [tex]\(150\)[/tex], and [tex]\(210\)[/tex]:
a) First, find the H.C.F. of [tex]\(120\)[/tex] and [tex]\(150\)[/tex]. It turns out that
[tex]\[
\text{H.C.F.}(120,150)=30.
\][/tex]
b) Next, find the H.C.F. of [tex]\(30\)[/tex] (the result above) and [tex]\(210\)[/tex]:
[tex]\[
\text{H.C.F.}(30,210)=30.
]
Thus, the overall H.C.F. is:
\[
\text{H.C.F.}(120,150,210)=30.
\][/tex]
According to the problem, this H.C.F. is expressed as:
[tex]\[
k^2-6=30.
\][/tex]
To find [tex]\(k\)[/tex], solve the equation:
[tex]\[
\begin{aligned}
k^2 - 6 &= 30,\\[1mm]
k^2 &= 30 + 6 = 36,\\[1mm]
k &= \sqrt{36}=6.
\end{aligned}
\][/tex]
So, the value of [tex]\(k\)[/tex] is [tex]\(6\)[/tex].
________________________________________
2. For the numbers [tex]\(17\)[/tex], [tex]\(23\)[/tex], and [tex]\(29\)[/tex]:
Since these are all prime numbers and do not have any common factors other than [tex]\(1\)[/tex]:
[tex]\[
\text{H.C.F.}(17,23,29)=1.
\][/tex]
________________________________________
Final Answers:
10. The value of [tex]\(k\)[/tex] is [tex]\(\boxed{6}\)[/tex].
11. The H.C.F. of [tex]\(17\)[/tex], [tex]\(23\)[/tex], and [tex]\(29\)[/tex] is [tex]\(\boxed{1}\)[/tex].
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