Menu Close

A-four-digit-number-has-the-following-properties-a-It-is-a-perfect-square-b-The-first-two-digits-are-equal-c-The-last-two-digits-are-equal-Find-all-such-numbers-




Question Number 13391 by Tinkutara last updated on 19/May/17
A four digit number has the following  properties:  (a) It is a perfect square  (b) The first two digits are equal  (c) The last two digits are equal.  Find all such numbers.
$$\mathrm{A}\:\mathrm{four}\:\mathrm{digit}\:\mathrm{number}\:\mathrm{has}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{properties}: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{It}\:\mathrm{is}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{The}\:\mathrm{first}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{are}\:\mathrm{equal} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{The}\:\mathrm{last}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{are}\:\mathrm{equal}. \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{such}\:\mathrm{numbers}. \\ $$
Answered by RasheedSindhi last updated on 19/May/17
let the number is  x+10x+100y+1000y  =11x+1100y  =11(x+100y)  Since the number is perfect  square it contains every prime  factor twice. So  11×11=121 is the divisor of  the required number.  Let the number is  121m^2   Since the number is of four  digits  121m^2 ≤9999                   11m<99  Hence the square root of the  number may be  11,22,33,...,99  On testing each of these  numbers only 88 is such number  whose square fulfills the  requirements  88^2 =7744
$$\mathrm{let}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is} \\ $$$$\mathrm{x}+\mathrm{10x}+\mathrm{100y}+\mathrm{1000y} \\ $$$$=\mathrm{11x}+\mathrm{1100y} \\ $$$$=\mathrm{11}\left(\mathrm{x}+\mathrm{100y}\right) \\ $$$$\mathrm{Since}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\mathrm{perfect} \\ $$$$\mathrm{square}\:\mathrm{it}\:\mathrm{contains}\:\mathrm{every}\:\mathrm{prime} \\ $$$$\mathrm{factor}\:\mathrm{twice}.\:\mathrm{So} \\ $$$$\mathrm{11}×\mathrm{11}=\mathrm{121}\:\mathrm{is}\:\mathrm{the}\:\mathrm{divisor}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{required}\:\mathrm{number}. \\ $$$$\mathrm{Let}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\:\mathrm{121m}^{\mathrm{2}} \\ $$$$\mathrm{Since}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\mathrm{of}\:\mathrm{four} \\ $$$$\mathrm{digits}\:\:\mathrm{121m}^{\mathrm{2}} \leqslant\mathrm{9999} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{11m}<\mathrm{99} \\ $$$$\mathrm{Hence}\:\mathrm{the}\:\mathrm{square}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{may}\:\mathrm{be} \\ $$$$\mathrm{11},\mathrm{22},\mathrm{33},…,\mathrm{99} \\ $$$$\mathrm{On}\:\mathrm{testing}\:\mathrm{each}\:\mathrm{of}\:\mathrm{these} \\ $$$$\mathrm{numbers}\:\mathrm{only}\:\mathrm{88}\:\mathrm{is}\:\mathrm{such}\:\mathrm{number} \\ $$$$\mathrm{whose}\:\mathrm{square}\:\mathrm{fulfills}\:\mathrm{the} \\ $$$$\mathrm{requirements} \\ $$$$\mathrm{88}^{\mathrm{2}} =\mathrm{7744} \\ $$
Commented by RasheedSindhi last updated on 19/May/17
It′s complete now.
$$\mathrm{It}'\mathrm{s}\:\mathrm{complete}\:\mathrm{now}. \\ $$
Commented by mrW1 last updated on 20/May/17
Great!
$${Great}! \\ $$
Commented by RasheedSindhi last updated on 20/May/17
 !1Wrm tol a sknahT^(↢)
$$\:\overset{\leftarrowtail} {!\mathrm{1Wrm}\:\mathrm{tol}\:\mathrm{a}\:\mathrm{sknahT}} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *