Determine the minimum value of a positive integer N such that:
o N has the form: 97.......97, and:
o N is divisible by 99
To be divisible by 99, the number must be divisible by 11 and by 9. Divisibility by 11 entails that the difference between the sum of the odd positions and the sum of the even positions must be a multiple of 11. Divisibility by 9 entails that the sum of all the digits must be a multiple of 9.
Each 97 in the representation of the number entails a difference of 2 in the respective modular sum, either up or down, and 2 is relatively prime to each of 11 and 9. Therefore, the number must consist of 99 repetitions of 97.
Verification:
syms n
n=sym(97);
for i=2:200
n=n*100+sym(97);
if mod(n,99) == 0
disp([i n])
end
end
shows the first two occurrences of divisibility by 99:
>> ninetysevenAndDivisibilityNuance
[99,
979797979797979797979797979797979797979797979797979797979797979797
979797979797979797979797979797979797979797979797979797979797979797
979797979797979797979797979797979797979797979797979797979797979797
]
[198,
979797979797979797979797979797979797979797979797979797979797979797
979797979797979797979797979797979797979797979797979797979797979797
979797979797979797979797979797979797979797979797979797979797979797
979797979797979797979797979797979797979797979797979797979797979797
979797979797979797979797979797979797979797979797979797979797979797
979797979797979797979797979797979797979797979797979797979797979797
]
>>
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Posted by Charlie
on 2021-12-12 08:42:40 |
solution
