Is 143 and 63 relatively prime numberd
1 answer:
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Answer:
Proofs are in the explantion.
Step-by-step explanation:
We are given the following:
1) for integer
.
1) for integer
.
a)
Proof:
We want to show .
So we have the two equations:
a-b=kn and c-d=mn and we want to show for some integer r that we have
(a+c)-(b+d)=rn. If we do that we would have shown that .
kn+mn = (a-b)+(c-d)
(k+m)n = a-b+ c-d
(k+m)n = (a+c)+(-b-d)
(k+m)n = (a+c)-(b+d)
k+m is is just an integer
So we found integer r such that (a+c)-(b+d)=rn.
Therefore, .
//
b) Proof:
We want to show .
So we have the two equations:
a-b=kn and c-d=mn and we want to show for some integer r that we have
(ac)-(bd)=tn. If we do that we would have shown that .
If a-b=kn, then a=b+kn.
If c-d=mn, then c=d+mn.
ac-bd = (b+kn)(d+mn)-bd
= bd+bmn+dkn+kmn^2-bd
= bmn+dkn+kmn^2
= n(bm+dk+kmn)
So the integer t such that (ac)-(bd)=tn is bm+dk+kmn.
Therefore, .
//
Answer:
77 m
Step-by-step explanation:
The geometry can be modeled by a right triangle. The distance from the viewpoint to the base of the tower is 110 m, the leg of the triangle adjacent to the angle. The height of the spire is the leg of the triangle opposite the given angle. The relevant trig relation is ...
Tan = Opposite/Adjacent
__
Using the given numbers, we find ...
tan(35°) = height/(110 m)
height = (110 m)tan(35°) = (110 m)×0.700208 ≈ 77.0 m
The height of the tower is about 77 meters.
