There is no three consecutive integers with same Parity of sum of its digitsLet D(a) denotes sum of digits of a in decimal. Examples D(49)=4+9=13Let P(a) denotes parity of a. Example P(2)=0 as even and P(3)=1 as odd.Questions: show that there is no a such thatP(D(a))=P(D(a+1))=P(D(a+2)).

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strasnihwhge · Accepted Answer

Step 1We will divide the numbers into two cases- Those ending with 9 and those which are not.Let, us select a number of the form (ending with 9)k9where k∈N. Clearly, the next number is of the form(k+1)0Now, k be either odd or even. Note, D(k9)=k+9 is of the opposite parity of k (Why?). And for the second number, D((k+1)0)=k+1+0=k+1 which is of opposite parity of k. So, if a number ends with 9, the number and its successive number are of same parity (0, 0) or (1, 1)Now, select a number of the form (not ending with 9)knwhere, k∈N and n∈{1,2,⋯,8}. The next number is of the formk(n+1)Clearly, D(kn)=k+n and D(k(n+1))=k+n+1which are of opposite parity.Now, see if the result you want follows.

There is no three consecutive integers with same

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