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Problem 4

The sum of two positive integers is 2022. The least common multiple of the numbers is 2018 times of their greatest common factor .

what is the least common multiple of the numbers?

两个数的和是2022.他们的最小公倍数是最大公约数的2018倍。

请问：他们的最小公倍数是多少？

Solution to Problem 3

answer：11

Solution：

let n equals the difference between Zoe’s and Chloe’s ages. The condition works when n has 9 factors. By inspection, 36 has 9 factors. Thus, Joey is 38 and will be 74 when Zoe is 37, so the answer is 7+4=11 .

解答：

假设n是田田和妈妈的年龄差，从题意得知n有9个因子。据观察得知36有9个因子。所以，爸爸今年38，当田田37岁的时候爸爸74岁。4+7=11.

Problem3

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe’s age will be an integral multiple of Zoe’s age. What will be the sum of the two digits of Joey’s age the next time his age is a multiple of Zoe’s age?

爸爸小朱、妈妈小敏和他们的女儿田田同一天生日。小朱比小敏大1岁，田田今天正好1岁。小敏有9个生日的年龄是田田年龄的整数倍，今天是第一个。请问：下一次小朱年龄是田田年龄整数倍的时候，父女俩的年龄和是多少？

Problem 2

Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?

小明有24个苹果。他想和小红、小华一起分这24个苹果，要求每人至少分到2个苹果，问有多少种分法？

Solution to Problem 1

answeCr（答案）：

Prisoner C shouted the color of his own hat.

犯人C喊出了他自己帽子的颜色

Solution：

If B and C were the same color, D would be able to deduce his color. Thus, when C doesn’t hear D shout his color, he knows he must be the opposite color as B.

解法：

如果B和C戴着同一种颜色的帽子，那么D就可以推导出自己帽子的颜色。C没有听见D喊，这就代表C戴着和B相反颜色的帽子。

Problem 1

In an act of kindness, King gave four prisoners the possibility of being released. He ordered the four of them to be buried to the neck, with only their heads sticking out of the earth. Looking ahead, unable to move, they see that there is a wall between prisoner A and B and know that the four are wearing hats – two white and two black – but they don’t know what color their own hat is.
Prisoner A and B see only their respective sides of the wall, C sees man B and D sees prisoners B and C. They have no communication with each other and will be taken back to jail within 10 minutes. . The only chance of being saved is if one of them accepts the color of the hat they are wearing. If he makes a mistake, they all return to prison. Minutes later, one of them screams and hits the color.
Which one shouted the color of his own hat?
How was he sure about the color of his hat?
国王给4个犯人一个被释放的机会。4个犯人脖子以下全被埋在泥土里，他们的头只能往前看，不能转动。犯人A和犯人B之间隔了一堵墙，所有犯人都知道他们中间有2个人戴黑帽子，2个人戴白帽子，但是大家都不知道自己帽子的颜色。
犯人A和犯人B只能看见墙，犯人C可以看见B，犯人D可以看见B和C。他们不能相互交流，10分钟以后他们会被送回监狱。他们唯一被释放的机会就是有人能正确喊出他自己帽子的颜色。如果他喊错，那么全体回监狱。几分钟后，其中一个人正确喊出了他自己帽子的颜色。
请问：谁喊对了帽子颜色？为什么？