What is the smallest positive multiple of 450 whose digits are all zeroes and ones? What is the smallest positive multiple of 450 whose digits are all zeroes and ones?

I tried guess and check but the numbers grew big fast.  Thanks in advance!
Can we agree that it must be an even multiple of 450?  Otherwise the last two digits will be 50.

What is the smallest positive multiple of 900 such that all the digits are 0s or 1s?

A rule of multiples of 9:  the sum of the digits of a multiple of 9 is a multiple of 9.

This rule goes both ways. If the sum of the digits is a multiple of 9, the number is a multiple of 9.

That makes  11111111100 our winner.
one more way. let x be the required number.

for a number to be multiple of 450, it should be multiple of 10.  so 10.y =x
now y should be multiple of 45. it should be first multiple of 5. so a number with only 1, o will be multiple of 5 if last digit is 0.
 so y = 2. w/9...dividing 10 by 5 gives 2 in numerator.

2.w should be mulitple of 9 and should contain only 1,0.

using the fact that sum of digits should be multiple of 9 for a number to be multiple of 9,

2.w should be 111111111.

adding two zeros extra.
11111111100.

thus 11111111100 is the required number.

lengthy process but easy to understand.