Hello all you maths experts!?

Question

Can you please help me on this question.

Find all four digit numbers which are a complete square and in which the first two digits are equal and the last two digits are also equal.

Thanks!!

Answer 1

The first poster is correct, here's one one way to get to the right answer.
If the first two digits are equal and the last two digits are also equal, the square must be divisible by 11. Since 11 is prime, its square root must be divisible by 11. Since the square has four digits, its square root must have two digits. So the square root must be a two digit number which is a multiple of 11. This narrows the search down to 11^2, 22^2, 33^2, 44^2, 55^2, 66^2, 77^2, 88^2, and 99^2. A quick calculator check finds the sole solution.

Answer 2

Let's first find all the possible last 2 digits.
Any square must end in 1,4,5,6 or 9.
So the only possible last 2 digits are
11,44,55,66 or 99.
Now any number ending in 11,55 or 99 = 3(mod 4),
so it is not a square.
Similarly, any number ending in 66 = 2(mod 4),
so it is not a square.
That leaves only 44 for the last 2 digits.
Now if we try all possibilities aa44,
we find that the only square is 7744 = 88².

Answer 3

Answer is 7744 arrived as under.

Let the number be denoted by aabb.
Then its value is 1100a + 11b = 11 ( 100a + b )

For this number to be a perfect square, the values of a and b should be such that 100a + b contains a factor of 11 and a square number. So values of a are taken from 1 to 9 and values of b ascertained such that 100a + b is divisible by 11. Eight such combinations are found. Of these only one combination a = 7 and b = 4 is such that 100a + b is left with a square quotient after dividing by 11.

Answer 4

The only one's I can think of are:

0, 0^2 = 0 = 0000 = aabb, where 0
and

88, 88^2 = 7744 = ccdd, where 0

Answer 5

sqrt(1100x+11y) is positive integer, dont it mean ?
I only find 7744 which is the root of 88.

I use this Maple program to do it
Kapix:=[0,0,0,0,0,0,0];
> Pacix:=1;
> for x from 1 to 9 do
> for y from 0 to 9 do
> if type(sqrt(1100*x+11*y),integer) then
> Kapix[Pacix]:=[1100*x+11*y,sqrt(1100*x+11*y)];
> Pacix:=Pacix+1;
> end if:
> end do:
> end do:
> Kapix;

use this URL to learn about Maple Programming
http://www.geocities.com/orichalc_of_moon/maple_intro.html

Answer 6

Don't forget 1111. I would be iffy on accepting 0000. In math, you write the number with the smallest number of digits for precision. Thus 00001 is written as 1

Answer 7

7744

sqrt 7744 = 88

88 x 88 = 7744

Answer 8

7744 which is 88*88

Answer 9

0000
7744