Answer :
To find the highest common factor (H.C.F.) of the numbers [tex]$144$[/tex], [tex]$216$[/tex], and [tex]$360$[/tex], we can use the following method:
1. First, find the H.C.F. of [tex]$144$[/tex] and [tex]$216$[/tex].
We use the Euclidean algorithm:
[tex]$$216 = 144 \times 1 + 72$$[/tex]
Now, we take the H.C.F. of [tex]$144$[/tex] and [tex]$72$[/tex]:
[tex]$$144 = 72 \times 2 + 0$$[/tex]
Since the remainder is [tex]$0$[/tex], the H.C.F. of [tex]$144$[/tex] and [tex]$216$[/tex] is [tex]$72$[/tex].
2. Next, find the H.C.F. of the result ([tex]$72$[/tex]) and the third number [tex]$360$[/tex].
Again, applying the Euclidean algorithm:
[tex]$$360 = 72 \times 5 + 0$$[/tex]
Since the remainder is [tex]$0$[/tex], the H.C.F. of [tex]$72$[/tex] and [tex]$360$[/tex] remains [tex]$72$[/tex].
Thus, the H.C.F. of [tex]$144$[/tex], [tex]$216$[/tex], and [tex]$360$[/tex] is:
[tex]$$\boxed{72}$$[/tex]
This corresponds to option D.
1. First, find the H.C.F. of [tex]$144$[/tex] and [tex]$216$[/tex].
We use the Euclidean algorithm:
[tex]$$216 = 144 \times 1 + 72$$[/tex]
Now, we take the H.C.F. of [tex]$144$[/tex] and [tex]$72$[/tex]:
[tex]$$144 = 72 \times 2 + 0$$[/tex]
Since the remainder is [tex]$0$[/tex], the H.C.F. of [tex]$144$[/tex] and [tex]$216$[/tex] is [tex]$72$[/tex].
2. Next, find the H.C.F. of the result ([tex]$72$[/tex]) and the third number [tex]$360$[/tex].
Again, applying the Euclidean algorithm:
[tex]$$360 = 72 \times 5 + 0$$[/tex]
Since the remainder is [tex]$0$[/tex], the H.C.F. of [tex]$72$[/tex] and [tex]$360$[/tex] remains [tex]$72$[/tex].
Thus, the H.C.F. of [tex]$144$[/tex], [tex]$216$[/tex], and [tex]$360$[/tex] is:
[tex]$$\boxed{72}$$[/tex]
This corresponds to option D.
