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Birth Dates Mystery

On John's 31st birthday his mother's age is 49. John adds the digits of his mother's age to get 4+9 = 13. He then repeats this process to get 1+3 = 4. Once he gets to a single digit he stops. He applies exactly the same process to his own age and gets 4 again.

He repeats this calculation for his next birthday, the birthday after that and so on. To his amazement, repeatedly adding the digits of his mother's age always gives the same value as repeatedly adding the digits of his own age. It seems that he and his mother share a special numerical relationship.

John then investigates whether he shares this numerical relationship with his father. On John's 31st birthday his father's age is 56. Adding the digits gives 5+6 = 11 and adding again gives 1+1= 2. Alas not! Furthermore, he discovers that he never does.

This age relationship between parent and child is not uncommon. What property must the parent's age have on the day the child is born if they are to share this numerical relationship?

Note by Anik Mandal
2 years, 8 months ago

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If the parent's age on the day the child is born is divisible by 9, the relationship will continue throughout the life. This property can be applied to all the number systems (decimal base 10, binary base 2, octal base 8, hex base 16) where the largest digit of the system is always one less than the base and the largest digit plays 'the role of 9'. Ali Baqar · 2 years, 8 months ago

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From the note we can easily conclude that john's mother was 18 when she gave birth to him.That implies that when John turned 1 his mother's age was 19.1+9=10 1+0=1=John's age.Whenever the mother's age is a multiple of 9 it satisfies the above condition. Adarsh Kumar · 2 years, 8 months ago

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This is definitely a nice relationship, but 31st birthday implies that john is 30 years old. you celebrate your first birthday when you are just born. so the number of birthdays is always one more than the age. Ansh Bhatt · 1 year, 9 months ago

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When age difference between two person is divisible by 9 then this numerical relationship comes true... Sajeeb Kumar · 2 years, 8 months ago

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Well could anyone prove that the numerical relationship will always hold whenever the mothers age is a multiple of 9.................. Anik Mandal · 2 years, 8 months ago

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@Anik Mandal A better solution:If one adds the digits of consecutive numbers then they form a sequence:1,2,3,4,5,6,7,8,9,1,2......Now,when a baby turns 1 his mother has to be 19,28,37...to satisfy the condition.I believe that much is enough. Adarsh Kumar · 2 years, 8 months ago

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@Anik Mandal First thing that needs to be noted here is that all the multiples of 9 have the same sum of digits and that is 9.Let us say that when a baby was born his/her mother was 18.Then when she will be 27 her son's/daughter's age will be 9.Same as the,sum of the digits of her age.Now let us say that when she gave birth she was 24 then she would be 33.Which doesn't satisfy the condition.CONCLUSION:when a baby turns 9 his/her mother's age's digit's sum should be 9 that is possible in the following cases:27,36,45,54,63,72,81,90,99.That implies:the mother has to be of an age that is a multiple of 9 when she gives birth. Adarsh Kumar · 2 years, 8 months ago

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The age difference between the child and parent should be a multiple of 9. In this case it is 18. Vihan Anand · 1 year, 8 months ago

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