Given the number 188188...188 (number 188 is written 101 times). Some digits of this number are crossed out. What is the largest multiple of 7, that could happen?
At first I tried a very much simpler version of the program below, trying each of the 303 possibilities of deleting just one digit. None was divisible by 7. A program similar to the one below, but not also deleting the first digit, tested every possibility of deleting two digits.
On a hunch I tried deleting the first digit and the last two digits. That was divisible by 7. That led to the below program, which varied which two others besides the first were divisible by 7. Indeed deleting the first three 1's works and gives the largest number available from deleting three digits, and it is divisible by 7, and that is the answer.
Playing around with intermediate values led me to the same conclusion as tomarken: no matter which three digits you remove, the result is divisible by 7.
a=repmat('188',101);
a=a(1,:);
digits 1000
syms n mx
n=mod(vpa(a),vpa(7))
pwr=1;
mx=0;
pmx=0; pmx2=0;
for p=2:302
for p2=p+1:303
b=a;
b(p2)=[];
b(p)=[];
b(1)=[];
n=mod(vpa(b),vpa(7));
if n==0
if b>=mx
mx=b;
pmx=p; pmx2=p2;
disp([1 pmx pmx2])
end
end
end
end
disp([pmx pmx2])
disp(mx)
>> erasedToMultOf7
n =
6.0 remainder from original number div by 7
1 2 3 |
1 2 4 |
1 3 4 | intermediate maxima
1 4 5 |
1 4 6 |
1 4 7 final maximum found removing these positions
4 7 positions removed other than first
8888881881881881881881881881881881881881881881881881881881881881
8818818818818818818818818818818818818818818818818818818818818818
8188188188188188188188188188188188188188188188188188188188188188
1881881881881881881881881881881881881881881881881881881881881881
88188188188188188188188188188188188188188188
>>
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Posted by Charlie
on 2021-04-21 10:39:46 |
computer solution
