(Originais Teachy 2024) - Question Hard of Mathematics

A store decides to reward its customers. Each customer will receive one of the six possible gifts available, according to their arrival order at the store. The gifts to be distributed are: a ball, a keychain, a pen, a soda, an ice cream, and a CD, in that order. The first customer at the store receives a ball, the second receives a keychain, the third receives a pen, the fourth receives a soda, the fifth receives an ice cream, and the sixth receives a CD. The seventh receives a ball, the eighth receives a keychain, and so on, according to the order of the gifts. The one-thousandth customer will receive as a gift a:

In Ana's lunchbox, she has 8 green grapes and 6 purple grapes. In João's lunchbox, he has 4 grapes of each color. If they combine all the grapes and divide them equally between them, who will have more grapes, Ana or João? Write two different addition equations that result in the same total number of grapes to help Ana and João divide the grapes fairly.

Imagine you have 432 candies and want to separate them into containers of 100, 10, and individual units to facilitate counting. How would you make this separation? Explain your answer by showing how the number 432 can be decomposed into powers of 10.

Consider the linear system below: 2x + 3y = 10 5x - 4y = -7 Transform this linear system into a matrix form using a matrix A and a column vector X, representing the unknowns. Write down the resulting matrix equation and inform what is the dimension of matrix A.