Problem 4. let m and n be two integers. show that m^3- n^3 is even if and only if m n is even.

The expression is even if both variables (m and n) are even or both are odd

Step-by-step explanation:

Let's remember the logical operations with even and odd numbers

odd*odd=odd

even*even=even

odd*even=even

odd-odd=even

even-even=even

even-odd=odd

Now, the original expression is:

which can be expressed as:

If m and n are both odd, then:

Then,

If m and n are both even, then:

Then,

Finally if one of them is even, for example m, and the other is odd, for example n, then:

Then,

In conclusion, the expression is even if both variables (m and n) are even or both are odd. If one of them is even and the other one is odd, then the expression is odd.

the value of x is 9 ⇒ answer b

step-by-step explanation:

* we can solve this problem using cosine and sine rule

- in δ abc

∵ ab = 36 , bc = 28 , ac = 16

- lets find the measure of angle b using the cosine rule

∵ cos b = [ab² + bc² - ac²]/2(ab)(bc)

∴ cos b = [36² + 28² - 16²]/2(36)(28)

∴ cos b = 19/21

∴ m∠b = cos^-1(19/21) ≅ 25°

- lets find the measure of angle a using the sine rule

∵ sina/bc = sinb/ac = sinc/ab

∴ sina/28 = sin25°/16 ⇒ by using cross multiplication

∴ sina = 28 × sin25°/16

∴ sina = 0.7396

∴ m∠a = sin^-1(0.7396) ≅ 48°

- from the figure m∠abd = m∠dbc

∵ m∠abc = 25°

∴ m∠abd = 25°/2 = 12.5°

- in δabd

∵ m∠a = 48° , m∠abd = 12.5°

∵ the sum of the measures of the interior angles of any δ is 180°

∴ m∠adb = 180° - (48° + 12.5°) = 119.5°

∵ ab = 36

∵ ad = x

- use the sine rule to find x

∵ sin∠adb/ab = sin∠abd/x

∴ sin119.5°/36 = sin12.5°/x ⇒ by using cross multiplication

∴ x = 36 × sin12.5°/sin119.5° = 8.95 ≅ 9

* the value of x is 9