4. Friends Petya, Vasya, and Kolya live in the same entrance. Climbing from the 1st to the 3rd floor, Petya passes a number of steps that is a multiple of four. Vasya, climbing from the 1st to the 6th floor, passes a number of steps that is a multiple of nine. How many steps will Kolya pass, who climbs from the 1st to the 9th floor, if the number of steps between floors is the same and less than $22?$
Answer: 144. Solution. From the 1st to the 3rd floor, there are 2 flights of stairs. According to the condition, the number of steps in these two flights is divisible by four. Then the number of steps in one flight is even. From the 1st to the 6th floor, there are 5 flights of stairs. According to the condition, the number of steps in these flights is divisible by nine. Then the number of steps in five flights is divisible by 18 (even and divisible by 9), and since 5 and 18 are coprime, the number of steps in one flight is divisible by 18. According to the condition, this number is less than 22, so it is exactly 18. Then the number of steps in eight flights from the 1st to the 9th floor is $18 \times 8=144$.

Grading criteria. Correct and justified solution - 7 points. It is shown that the number of steps in a flight of stairs is an even number or that it is divisible by 9 - 2 points. It is proven that the number of steps in a flight of stairs is even and divisible by 9 - 3 points. It is proven that the number of steps in a flight of stairs is divisible by 18 - 4 points. In other cases - 0 points.