Generations apart
Number Theory
Level
3
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
Problem boils down to: Find two consecutive prime numbers with a gap of >6, with 6 consecutive factors
Prime numbers<100 with gap >=6 23,31,47,53,61,73,83,89
All candidates except for 89 are exactly 6 gaps thus they must start with 1 as the age of the grandson since the first year will be a prime for grandma.
e.g. 31, 31/1=31,32/2=16,33/3=11,34/4=8.5 FAIL
only 61 passes for the 6 gaps <100.
for 89. The series must contain either 91 or 94. 91 has factors 7,13 and 94 has factors 2,47. 92/8=11.5 FAIL, 92/14=6.57.. FAIL. 95/3=31.66 FAIL.
Only series that passes <100 is that starting with 61 and with factors starting at 1.
i.e. 61,62,63,64,65,66 1,2,3,4,5,6
Answer: Her final age is 66.
This is the only answer for <100