After the preschool fee of $330, Cody paid an additional 770 – 330 = $440 for after-school care. Dividing $440 by the hourly rate of $5.50, we find that Cody’s son must have spent 80 hours in after-school care. 

 Let n – 1, n and n + 1 be positive consecutive integers such that (n – 1) ×n × (n + 1) = 16 × [(n – 1) + n + (n + 1)]. Simplifying and solving for n, we have n(n² – 1) = 16 × 3n → n³ – n = 48n → n³ – 49n = 0

=> n(n² – 49) = 0. So, n = 0 or n² − 49 = 0 → (n + 7)(n – 7) = 0 => n = −7 or n = 7. Since n is a positive integer, it follows that the three consecutive numbers are 6, 7 and 8. The difference between their product and sum is 6 × 7 × 8 – (6 + 7 + 8) = 336 – 21 = 315. 

The ratio of the surface area of Wilbur’s plane to the surface area of Orville’s plane is the square of the scale factor of Wilbur’s mini replica of Orville’s plane. Since Wilbur’s mini replica has linear dimensions that are \(\dfrac{1}{2}\) the size of Orville’s model airplane, it follows that the ratio between the lift forces on Wilbur’s and Orville’s planes is \(\dfrac{1}{2}\times\dfrac{1}{2}=\dfrac{1}{4}\) 

If we add the wins of both Flying Turtles and Dolphins, we have to then subtract the 19 wins for the games they played against each other  regardless of who won those games. The value of F + D, then, is 95 + 84 – 19 = 160. 

Let x, y be the length and breadth of a rectangle and 16 is the length of diagonal as shown below:

Using the Pythagorean Theorem in the right triangle formed by sides with the diagonal, we get: x² + y² = 256²

=> \(y=\sqrt{256-x^2}\) and \(A\left(x\right)=x\sqrt{256-x^2}\)

that is the function we have to maximize. Notice that x can vary between x = 0 (the rectangle collapses to a vertical line) and x = 16 (the rectangle collapses to a horizontal line), and in both extremes the area is 0. Let’s find critical points for A:

\(0=A'\left(x\right)=\sqrt{256-x^2}-x-\dfrac{x}{\sqrt{256-x^2}}\\ \Rightarrow\dfrac{x^2}{\sqrt{256-x^2}}=\sqrt{16-x^2}\\ \Rightarrow x^2=256-x^2\)

and so \(x=\sqrt{128}\) is the only critical value

Moreover, A(\(\sqrt{8}\)) = 8 > 8 so it is the maximum

A.BC + D.EF = 0.13 + 0.24 = 0.37

After the first spin, the probability that the pointer lands on the same color is: 1 : 3 = 1/3

We have: 110011₂ = 1 x 2⁵ + 1 x 2⁴ + 1 x 2¹ + 1 x 2⁰ = 51₁₀ =  6 x 8¹+ 3 x 8⁰ = 63₈

So 110011₂ is rewritten as 63₈ in octal (base 8)

It is not wrong. Just because of your OLD phone/computer

An amicable (or friend) pair  (m,n)  consists of two integers  m,n  for which the sum of proper divisors (the divisors excluding the number itself) of one number equals the other. Amicable pairs are occasionally called friendly pairs (Hoffman 1998, p. 45), although this nomenclature is to be discouraged since the numbers more commonly known as friendly pairs are defined by a different, albeit related, criterion. Symbolically, amicable pairs satisfy

s_(m) = n
(1)
s_(n) = m
(2)

where

s_(n)=sigma_(n)-n
(3)

is the restricted divisor function. Equivalently, an amicable pair  [(m,n)]  satisfies

sigma_(m)=sigma_(n)=s_(m)+s_(n)=m+n
(4)

where  sigma_(n)  is the divisor function. The smallest amicable pair is (220, 284) which has factorizations

220 = 11·5·2²
(5)
284 = 71·2²
(6)

giving restricted divisor functions

s₍₂₂₀₎ = sum{1,2,4,5,10,11,20,22,44,55,110}
(7)= 284
(8)
s₍₂₈₄₎ = sum{1,2,4,71,142}
(9)
= 220.
(10)

The quantity

sigma_(m)=sigma_(n)=s_(m)+s_(n),
(11)

in this case,  220+284=504 , is called the pair sum. The first few amicable pairs are (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), ... (OEIS A002025 and A002046). An exhaustive tabulation is maintained by D. Moews.

Still some more, but with "x", "/", ...

123-45-67+89=100

1 + 2 + 3 – 4 + 5 + 6 + 78 + 9 = 100
123 +45 - 67 + 8 - 9 = 100

123 + 4- 5 + 67 - 89 = 100
 

An amicable pair  [(m,n)]  consists of two integers  [m,n]  for which the sum of proper divisors (the divisors excluding the number itself) of one number equals the other. Amicable pairs are occasionally called friendly pairs (Hoffman 1998, p. 45), although this nomenclature is to be discouraged since the numbers more commonly known as friendly pairs are defined by a different, albeit related, criterion. Symbolically, amicable pairs satisfy

[s(m)] [=] [n]
(1)
[s(n)] [=] [m,]
(2)

where

[s(n)=sigma(n)-n]
(3)

is the restricted divisor function. Equivalently, an amicable pair  [(m,n)]  satisfies

[sigma(m)=sigma(n)=s(m)+s(n)=m+n,]
(4)

where  [sigma(n)]  is the divisor function. The smallest amicable pair is (220, 284) which has factorizations

[220] [=] [11·5·2^2]
(5)
[284] [=] [71·2^2]
(6)

giving restricted divisor functions

[s(220)] [=] [sum{1,2,4,5,10,11,20,22,44,55,110}]
(7)
[=] [284]
(8)
[s(284)] [=] [sum{1,2,4,71,142}]
(9)
[=] [220.]
(10)

The quantity

[sigma(m)=sigma(n)=s(m)+s(n),]
(11)

in this case,  [220+284=504] , is called the pair sum. The first few amicable pairs are (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), ... (OEIS A002025 and A002046). An exhaustive tabulation is maintained by D. Moews.

Why don't you post some?

An amicable pair  [(m,n)]  consists of two integers  [m,n]  for which the sum of proper divisors (the divisors excluding the number itself) of one number equals the other. Amicable pairs are occasionally called friendly pairs (Hoffman 1998, p. 45), although this nomenclature is to be discouraged since the numbers more commonly known as friendly pairs are defined by a different, albeit related, criterion. Symbolically, amicable pairs satisfy

[s(m)] [=] [n]
(1)
[s(n)] [=] [m,]
(2)

where

[s(n)=sigma(n)-n]
(3)

is the restricted divisor function. Equivalently, an amicable pair  [(m,n)]  satisfies

[sigma(m)=sigma(n)=s(m)+s(n)=m+n,]
(4)

where  [sigma(n)]  is the divisor function. The smallest amicable pair is (220, 284) which has factorizations

[220] [=] [11·5·2^2]
(5)
[284] [=] [71·2^2]
(6)

giving restricted divisor functions

[s(220)] [=] [sum{1,2,4,5,10,11,20,22,44,55,110}]
(7)
[=] [284]
(8)
[s(284)] [=] [sum{1,2,4,71,142}]
(9)
[=] [220.]
(10)

The quantity

[sigma(m)=sigma(n)=s(m)+s(n),]
(11)

in this case,  [220+284=504] , is called the pair sum. The first few amicable pairs are (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), ... (OEIS A002025 and A002046). An exhaustive tabulation is maintained by D. Moews.

Your answer is incorrect, try again

SORRY, the answer is incorrect, but I have made a mistake when for him

You can see the answer at: http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/15puzzle.pdf

P/s: You always type incorrectly

Ex: Standand, not stardard

The probablity of rolling one first time is \(\dfrac{1}{6}\)

Six times that, we have: \(\left(\dfrac{1}{6}\right)^6\)=\(\dfrac{1}{46656}\)

But is this correct, because it is BEST IQ, right?