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Math Help - How many sq. numbers are there?

  1. #1
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    How many sq. numbers are there?

    Hello,

    Please help me find solution to this.

    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.
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  2. #2
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    Quote Originally Posted by skyskiers View Post
    Hello,

    Please help me find solution to this.

    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
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  3. #3
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    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 ....
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  4. #4
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    Quote Originally Posted by Bingk View Post
    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 ....
    How about this approach?

    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.
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  5. #5
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    Quote Originally Posted by CaptainBlack View Post
    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?
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  6. #6
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    Quote Originally Posted by skyskiers View Post
    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
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  7. #7
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    Quote Originally Posted by CaptainBlack View Post
    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.
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