Help with maths explanation and explaining answers.?

Question

Hi, I've asked the question below last week and have received fantastic results, but I just don't understand them.

Prove that 43 ^ n + 83 x 92 ^ 3n – 1 is divisible by 7 for all positive integers n,

Can you please help by either:

Defining the following two answers into simpler terms, enough for a young teenager to understand:


Reduce mod 7: 43^n= 1^n=1mod7
83=6mod7 and 92^(3n-1) =(13*7+1)^(3n-1)=1^(3n-1)=1mod...

Plug these in : 1 + 6(1) =7mod 7. Therefore your expression
is divisible by 7 for all n.

or


43 to any power mod 7 always =1
92 to any power mod 7 always =1
83 mod 7 =6
so 1+1*6 is the mod of the whole expression for any n. Which is divisible by 7.

OR

Answer the question with a new answer, making it as simple as possible. Thanks.

Answer 1

OK, I'll try.

First, observe that 43 - 1 = 42 = 6 * 7, so that 7 divides 43 -1. In congruence language,this is equivalent to saying that 43 = 1 (mod 7) (43 is congruent to 1 module 7, here = means congruent to). We know that if a =b (mod k), then a^n = b^n (mod k), n positive integer. This a property of congruences. Therefore,

43^n = 1^n = 1 (mod 7).

Now, 83 - 6 = 77 = 11 * 7, so that 83 = 6 mod(7)

92 -1 = 13 * 7, so that 92 = 1 (mod 7), This implies
92 ^(3n -1) = 1^(3n -1) =1 (mod 7). Then we have

83 = 6 mod(7) and
92 ^(3n -1) =1 (mod 7). According to the properties of congruences, we can multiply 2 congruences of the same module, getting:

83 * 92 ^(3n -1) = 6 * 1 = 6 (mod 7)., that is
83 * 92 ^(3n -1) = 6 (mod) 7
And we have seen that
43^n = 1 (mod 7). According to the properties of congruences, we can add congruences of the same module, so that
43^(n) + 83 * 92 ^(3n -1) = 6 ¨+ 1 (mod 7), that is
43^(n) + 83 * 92 ^(3n -1) = 7(mod 7). This implies that

43^(n) + 83 * 92 ^(3n -1) - 7 is divisible by 7 and therefore , since 7 divides 7, 43 ^ n + 83 x 92 ^ (3n – 1) is divisible by 7.

Answer 2

You can go there :
http://en.wikipedia.org/wiki/Modular_arithmetic
for a primer on modular arithmetic.
Basically, if the division of a1 by n gives you a remainder of b1
and the division of a2 by n gives you a remainder of b2, then the division of a1a2 by n gives you a remainder of b1b2.
You have the same relation for addition : remainder of the division of (a1+a2) by n is (b1+b2)
By the same rationale, the remainder of the division of a1^p by n is b1^p.

so... 43=7*6 +1. the remainder is 1.
The remainder of the division of 43^n by 7 is then 1^n = 1

92 = 7*13 + 1. The remainder is 1
The remainder of the division of 92^(3n-1) by 7 is 1^(3n-1) = 1

Now use the addition rule:
the remainder of the division of 43 ^ n + 83 x 92 ^ (3n-1) by 7 is the remainder of the division of 1 + 83*1 by 7
1+83*1 = 84
84 = 7*12
the remainder is zero. Your function is divisible by 7.

Answer 3

a million.First tin can has quantity of five*4*15= 3 hundred cm^3. Cylinder has quantity pi*(7/2)^2*10 = 384.845 approximately. So, cylinder has extra advantageous skill with the aid of 384.845-3 hundred=80 4.845 cm^3 approximately. 2.i) SO, 2pi*r*5=ninety 4.2. SO, r=ninety 4.2/(2*pi*5) = 3 ii)V=3.14*3^2*5 = 141.3 cm^3