Prove that for each natural number n > 43, we can write n = 6xn + 9yn + 20zn for some nonnegative integers xn, yn, zn. then prove that 43 cannot be written in this form. (hint: write 44, 45, 46, 47, 48, and 49 in the given form. use induction to prove that any larger number can be written in the given form.)

Jonathan pays $1.90 per pound for potatoes. he buys 8.3 pounds of potatoes. he determines that he will pay $15.77, before tax, for the potatoes. which best describes the reasonableness of jonathan’s solution? jonathan’s answer is reasonable because there are two decimal places in the factors and two in the product. jonathan’s answer is unreasonable because there are three decimal places in the factors and two in the product. jonathan’s answer is reasonable because is 16, and 16 is close to 15.77. jonathan’s answer is unreasonable because is 8, and 8 is not very close to 15.77.

Colby and jaquan are growing bacteria in an experiment in a laboratory. colby starts with 50 bacteria in his culture and the number of bacteria doubles every 2 hours. jaquan starts with 80 of a different type of bacteria that doubles every 3 hours. let x equal number of days. colbys experiment follows the model: a.) y=50*2^x b.) y=50*2^8x c.) y=50*2^12x jaquans experiment follows the model: a.)80*2^x b.)80*2^8x c.)80*2^12x

Bo is buying a board game that usually costs bb dollars. the game is on sale, and the price has been reduced by 18\%18%. what could you find the answer?