Thread: How many sq. numbers are there?

1. How many sq. numbers are there?

Hello,

A 4 digit number abcd ( all distinct digits) is a perfect square such that its reciprocal dcba is also a perfect sq. and dcba is a factor of abcd? How many such numbers are there?

Any pointers well be appreciated.

2. Originally Posted by skyskiers
Hello,

A 4 digit number abcd ( all distinct digits) is a perfect square such that its reciprocal dcba is also a perfect sq. and dcba is a factor of abcd? How many such numbers are there?

Any pointers well be appreciated.
Each of the numbers is the square of a number in [32,99]. If we assume that abcd>=dcba then if abcd=uv^2 and dcba=wx^2 then wx|uv.

Also since there are no palindromic squares in the given range we know that wx<uv. This reduces the candidates for wx to the range [32,44] which is small enough to be checked by exhaustive search.

CB

3. Um ... the way I did it is tedious. Basically, since abcd is a 4 digit number with distinct digits, I considered the set of all 4 digit perfect squares. We get that from $32^2$ to $99^2$ gives us this set.

From that set, we remove the ones with non-distinct digits, and we get the following list of numbers

1024,1089,1296,1369,1764,1849,1936,2304,2401,2601, 2704,2809,2916,3025,
3249,3481,3721,4096,4356,4761,5041,5184,5329,5476, 6084,6241,6724,7056,
7396,7569,7921,8649,9025,9216,9604,9801

We get the following pair:
1089,9801

Also, 9 * 1089 = 9801

* We are fortunate that none of the listed numbers ends in zero (i.e. d = 0), otherwise we would also have to consider 3 digit perfect squares, which would add more work ....

4. Originally Posted by Bingk
Um ... the way I did it is tedious. Basically, since abcd is a 4 digit number with distinct digits, I considered the set of all 4 digit perfect squares. We get that from $32^2$ to $99^2$ gives us this set.

From that set, we remove the ones with non-distinct digits, and we get the following list of numbers

1024,1089,1296,1369,1764,1849,1936,2304,2401,2601, 2704,2809,2916,3025,
3249,3481,3721,4096,4356,4761,5041,5184,5329,5476, 6084,6241,6724,7056,
7396,7569,7921,8649,9025,9216,9604,9801

We get the following pair:
1089,9801

Also, 9 * 1089 = 9801

* We are fortunate that none of the listed numbers ends in zero (i.e. d = 0), otherwise we would also have to consider 3 digit perfect squares, which would add more work ....

Since abcd and dcba are both sq. nos. and abcd is a factor of dcba, it follows that dcba/abcd should be a square integer.

Since both abcd and dcba are 4 digit nos., dcba/abcd =n should be either one of 1,4,9. Obviously n can not be 1 as that would both the nos. equal. So n can be 4 or 9.

a < 4 , because if a >=4 and n =4 or 9, dcba will become a 5 digit no.
Now, as all the perfect sqs. end in 1,4,9,6,0,5, it follows that a has to be 1.

Now, since we have been able to settle the scores with a, it follows that n != 4 ( because no multiple of 4 ends in 1). So n =9.

Therefore, d=9 and a=1

Now something like this should work,

1000*d + 100c +10b +a = 9 * [ 1000 *a + 100b + 10c +d]

1000*9 + 100c +10b +1 = 9 * [ 1000 *1 + 100b + 10c +9]

9000 +100c+ 10b = 9[1000 + 100b +10c +9 ]

9001 +100c+ 10b = 9000 +900b +90c +81

-80 = -10c + 890b

if b=0, c comes 8. No other single valued of b and c satisfies this eqn.

Therefore, dcba = 9801 and abcd 1089

I know this is a bit longer approach but much of this should be mental maths, which i took hours to come to.

5. Originally Posted by CaptainBlack
Each of the numbers is the square of a number in [32,99]. If we assume that abcd>=dcba then if abcd=uv^2 and dcba=wx^2 then wx|uv.

Also since there are no palindromic squares in the given range we know that wx<uv. This reduces the candidates for wx to the range [32,44] which is small enough to be checked by exhaustive search.

CB
Thanks for your input. Can you please clarify a bit more on how did you filter the nos. out to 44?

6. Originally Posted by skyskiers
Thanks for your input. Can you please clarify a bit more on how did you filter the nos. out to 44?
If wx is a proper divisor or uv it is less than a half of uv. However uv<=99 so wx<=44.

CB

7. Originally Posted by CaptainBlack
If wx is a proper divisor or uv it is less than a half of uv. However uv<=99 so wx<=44.

CB
Thanks...didnt think of that one.