Generation gap
Number Theory
Level
pending
A woman and her grandson have the same birthday. For six consecutive birthdays, she is an integral multiple of her grandson's age. How old is the grandmother at the sixth of these birthdays?
The answer is 66.
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1 solution
Looking to the table of prime, from 30 to 113 there is no gap of six between adjacent primes. So the first age of the child must be 1 so that any number can be its multiple. So the six birthdays of the child must be at the age of 1, 2, 3, 4, 5, 6. And the range of her age must have five unit gaps. 1st birth day she must have a prime as her age so that 1 divides what ever her age. Her sixth birth day have to be multiple of 6. Since there are only 5 gaps, her seventh birth day must be a prime number. So her age is between prime and prime, include the first prime, exclude the second..
31 to 36 is good, but the fourth birth days 4 does not divide 34.
From 37 to 47 there is no sufficient gap.
47 to 53 and 53 to 59, her 5th birth day is not a multiple of that of the child.
59 to 61 not sufficient gap.
61 to 66:- 1|61, 2|62, 3|63, 4|64, 5\65, and 6|66. Her age on sixth birthday is 66 a reasonable possibility.