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Drupady [299]
3 years ago
15

What is the remainder when 3^99 is divided by 8?

Mathematics
2 answers:
yaroslaw [1]3 years ago
8 0
3^99 can be re-written as (4-1)^99
Let’s call 4 as x,and -1as y.

When we do the binomial expansion of (x+y)^99, and temporarily omit the coefficient, we obtain,
X^99+x^98*y+x^97*y^2+x^96*y^3.......x*y^98+y^99.

We know that 4 raised to any natural number grater than one,will divide 8 completely (4^2=16=8*2, and therefore every subsequent power of 4 can be rewritten in the form of 8k, where k is a natural number.)

The last two terms of this binomial expansion are,
(99 c 98) *x*y^98 + y^99.....(x C y represents x “choose” y)
99 C 98 is nothing but 99 ....... ( by using the n C r formula )
So the last two terms will be,
99*4*1+(-1)
=396-1=395
395 mod 8= (392 mod 8) + (3 mod 8) = 0+3=3
So our required answer has been obtained: 3

Hope that helps
Gnom [1K]3 years ago
3 0

Answer:

3

Step-by-step explanation:

3^1=3 divided by 8 will give a remainder of 3

3^2=9 divided by 8 will give a remainder of 1

3^3=27 divided by 8 will give a remainder of 3

3^4=81 divided by 8 will give a remainder of 1

3^5=243 divided by 8 will give a remainder of 3

You will notice that 3^n will give 3 when n is odd number and 1 when n is even number

Since n=99 and 99 is an odd number,

Then the remainder is 3

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In RSM camp, there are four counselors for every fifty seven students and three teachers for every five counselors.
Ray Of Light [21]

Answer:

Students : Counselors : Teachers = 285: 20 : 12

Step-by-step explanation:

In the camp, the ratio of:

( C: S )  Counselors : Students   is  4   :  57

( T: C )  Teachers :   Counselors  is  3   :  5

To determine the ratio of   -  Students : Counselors : Teachers

Here given      -  C : S  = 4 : 57

Multiplying both sides by 5, we get

C: S  = 4 x 5  : 57 x 5  = 20 : 285

So, for every 20 counselors, there are 285 students      ...... (1)

Similarly, given   Teachers :   Counselors  is  3   :  5

Multiplying both sides by 4, we get

T  : C  = 3 x 4  : 5 x 4  = 12  : 20

So, for every 12  teachers, there are 20 Counselors      ...... (2)

Hence, in both equation 1 and 2 the number of counselors is same = 20 .

Hence, both the ratios acne be combined.

So we get C : S  = 20 : 285      and T : C  = 12: 20

Inverting the values

⇒  S : C  = 285 : 20         and C : T  = 20 : 12

or, S : C : T  = 285: 20 : 12

or, Students : Counselors : Teachers = 285: 20 : 12

Hence, for every 285 students there are 20 counselors and 12 teachers.

4 0
3 years ago
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