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alisha [4.7K]
3 years ago
7

Trisha is going to make a border of icing around a circular cake with a diameter of 8 inches which of the following represents t

he amount of icing Trisha will need? A.12.56, B.25.12, C.18.62, D.50.24
Mathematics
1 answer:
nadya68 [22]3 years ago
4 0

Answer:

B. 25.12

Step-by-step explanation:

We are simply finding the perimeter of the cake.

Perimeter: π × diameter

Perimeter: 3.14 × 8

Perimeter: 25.12 inches

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Find a and b such that gcd(a, b) = 14, a &gt; 2000, b &gt; 2000, and the only prime divisors of a and b are 2 and 7.
choli [55]

Answer:

a = 2^\alpha.14 \ ,\alpha\geq 8\\b = 7^\delta.14 \ , \delta\geq 3

Step-by-step explanation:

As 2  and 7 are the only prime divisors of both a and b we know that both can be written as:

a = 2^\alpha . 7^\beta \ , \alpha ,\beta\ \in \mathbb{N}_{0}\\b = 2^\gamma . 7^\delta \ , \gamma ,\delta\ \in \mathbb{N}_{0}\\

Where \mathbb{N}_{0} is the set of all natural numbers adding the zero (careful because this part is important as I'll explain next).

We also know that 14 divides both numbers and that is actually the greatest common divisor between them. So we can rewrite a and b as follows:

a = 2^\alpha . 7^\beta.14 \ , \alpha ,\beta\ \in \mathbb{N}_{0}\\b = 2^\gamma . 7^\delta.14 \ , \gamma ,\delta\ \in \mathbb{N}_{0}\\

Why do I write them like this? Because this way is easier to observe that if \alpha and \gamma were both greater than zero, then 28 would divide both hence 14 wouldn't be their g.c.d.. Likewise, if \beta and \delta were both greater than zero, then 98 would divide both and once again, 14 wouldn't be their g.c.d.

So either of them has to be equal to zero. And then we have that

a = 2^\alpha.14 \\b = 7^\delta.14

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