how many positive integers less than 400 are divisible by both 7 and 11?

how many perfect squares less than 1000 have a ones digit of 2, 3 or 4?

a solar system contains four planets and a sun as shown. planet A completes an orbit around the sun in 20 months; planet B completes an orbit in 36 months; planet C completes an orbit in 40 months; and planet D completes an orbit in 48 months. If the four planets lie along a straight line in May of the year 2000, what is the next year that they will again lie along the same line? Assume that the orbits are circular and that a year has exactly 12 months.

Guest Sep 15, 2022

#1**0 **

1) If it is divisible by 7 and 11, it must be divisible by \(\text{lcm}(7, 11) = 77\).

Can you take it from here?

BuilderBoi Sep 15, 2022

#2**0 **

2) No number can be squared and have a ones digit of 2 or 3.

The only way it can have a last digit of 4 is if the number ends in 2 (2, 12, and 22).

So there are \(\color{brown}\boxed{3}\) numbers that work.

BuilderBoi Sep 15, 2022