I don\'t understant why the perfect square is must be a multiple of both 3 and 9

Question

I don't understant why the perfect square is must be a multiple of both 3 and 9?

Problem 3. Can we put the numbers 1,2,3,. 2019 in a row to form one number which is a perfect square? Proof. What all the numbers that we can form have in common All the possible numbers we can form have equal sum of digits 1+2+34 +2019 =-2019 * 2020-2039 190. This number is a multiple of 3, but not a multiple for 9. However any perfect square which is a multiple for three is also a multiple of 9. So we cannot form a perfect square

Explanation / Answer

Note that in the prime factorisation of a perfect square, All the powers of the prime factors must be a even number.

for example , Consider 20^2 = ( 5 x 2^2 )^2

= (5^2 x 2^4) , Powers of 5 and 2 are even.

Now If the number is multiple of 3 , But not the multiple of 9, That means in the prime factorisation of that number , 3 appears only once. i.e., the power of 3 is 1.  

As We have seen already , A number is a perfect square, If and only if powers of all the prime factors in the prime factorisation of the number are even. But Here We have the power of 3 as 1 which is odd. Hence the number can't be perfect square.

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