Study the partial work shown for the following division problem.

The correct options include B "The sine function rises in the intervals (0°, 90°) and (270°, 360°)." E "Both sinusoidal functions reach a peak value of 1." and F "Both functions exhibit periodic behavior."

 I divided by 100 and shifted the decimal point two places left.

The profit produced by combining sales of gold label and black label equals the total of their individual profits. From the table you find the gold label's profit and similarly from another table find the black label's profit . Adding these yields . Thus, option 5 is the correct choice.

(a) The multiplicative inverse of 1234 (mod 4321) is x so that 1234*x ≡ 1 (mod 4321). We can apply Euclid's algorithm:

4321 = 1234 * 3 + 619

1234 = 619 * 1 + 615

619 = 615 * 1 + 4

615 = 4 * 153 + 3

4 = 3 * 1 + 1

Now we will express 1 as a linear combination of 4321 and 1234:

1 = 4 - 3

1 = 4 - (615 - 4 * 153) = 4 * 154 - 615

1 = 619 * 154 - 155 * (1234 - 619) = 619 * 309 - 155 * 1234

1 = (4321 - 1234 * 3) * 309 - 155 * 1234 = 4321 * 309 - 1082 * 1234

This reduces to

1 ≡ -1082 * 1234 (mod 4321)

Thus, the inverse is

-1082 ≡ 3239 (mod 4321)

(b) Since both 24140 and 40902 are even, their GCD cannot equal 1, indicating no inverse exists.

Answer: 11, 13

Explanation:

We need to identify two consecutive integers whose product is 143. If we let x represent the larger of the two integers, the smaller integer can be expressed as (x-2). Their product equals 143, which gives us

By solving the equation:

this results in a second-order equation with two real roots: x=13 and x=-11. We disregard the negative outcome, leaving us with x=13. Hence, the two numbers are

x=13

x=13-2=11