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

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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.

Sep 15, 2022

#1
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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?

Sep 15, 2022
#2
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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.

Sep 15, 2022
#3
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3) find the lcm of 20, 36, 40, and 48.

You should get it from there.

Sep 15, 2022
edited by BuilderBoi  Sep 15, 2022
#4
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Number 2) is incomplete in the above answer.

A square also can end in 4 if its root ends in 8.

Namely:    8,  18,  28

So, counting the three that end in 2, there are a total of 6.

.

Sep 16, 2022