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A-two-digit-number-has-the-property-that-the-square-of-its-tens-digit-plus-ten-times-its-units-digit-is-equal-to-the-square-of-its-units-digit-plus-ten-times-its-tens-digit-Find-all-two-digit-numbers

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Question Number 16935 by Tinkutara last updated on 28/Jun/17
A two-digit number has the property that the square of its tens digit plus ten times its units digit is equal to the square of its units digit plus ten times its tens digit. Find all two digit numbers which have this property, and are prime numbers.
$$\mathrm{A}\:\mathrm{two}-\mathrm{digit}\:\mathrm{number}\:\mathrm{has}\:\mathrm{the}\:\mathrm{property} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\:\mathrm{its}\:\mathrm{tens}\:\mathrm{digit}\:\mathrm{plus} \\ $$$$\mathrm{ten}\:\mathrm{times}\:\mathrm{its}\:\mathrm{units}\:\mathrm{digit}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{square}\:\mathrm{of}\:\mathrm{its}\:\mathrm{units}\:\mathrm{digit}\:\mathrm{plus}\:\mathrm{ten}\:\mathrm{times} \\ $$$$\mathrm{its}\:\mathrm{tens}\:\mathrm{digit}.\:\mathrm{Find}\:\mathrm{all}\:\mathrm{two}\:\mathrm{digit} \\ $$$$\mathrm{numbers}\:\mathrm{which}\:\mathrm{have}\:\mathrm{this}\:\mathrm{property},\:\mathrm{and} \\ $$$$\mathrm{are}\:\mathrm{prime}\:\mathrm{numbers}. \\ $$
Answered by RasheedSoomro last updated on 28/Jun/17
Let u is unit digit and t is tens digit. t^2 +10u=u^2 +10t t^2 −u^2 +10u−10t (t−u)(t+u)−10(t−u)=0 (t−u)(t+u−10)=0 t=u ∣ t=10−u As the number is prime u≠2,4,5,6,8 u may be 1,3,7 & 9 If t=u the numbers are 11,33^(×) ,77^(×) ,99^(×) If t=10−u the numbers may be 91^(×) ,73,37,19 Total such numbers which have mentioned property: 11,33,77,99,91,73,37,19 Among which primes are: 11,73,37,19 (Four)
$$\mathrm{Let}\:\mathrm{u}\:\mathrm{is}\:\mathrm{unit}\:\mathrm{digit}\:\mathrm{and}\:\mathrm{t}\:\mathrm{is}\:\mathrm{tens}\:\mathrm{digit}. \\ $$$$\mathrm{t}^{\mathrm{2}} +\mathrm{10u}=\mathrm{u}^{\mathrm{2}} +\mathrm{10t} \\ $$$$\mathrm{t}^{\mathrm{2}} −\mathrm{u}^{\mathrm{2}} +\mathrm{10u}−\mathrm{10t} \\ $$$$\left(\mathrm{t}−\mathrm{u}\right)\left(\mathrm{t}+\mathrm{u}\right)−\mathrm{10}\left(\mathrm{t}−\mathrm{u}\right)=\mathrm{0} \\ $$$$\left(\mathrm{t}−\mathrm{u}\right)\left(\mathrm{t}+\mathrm{u}−\mathrm{10}\right)=\mathrm{0} \\ $$$$\mathrm{t}=\mathrm{u}\:\:\mid\:\:\mathrm{t}=\mathrm{10}−\mathrm{u} \\ $$$$\mathrm{As}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\mathrm{prime} \\ $$$$\mathrm{u}\neq\mathrm{2},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{8} \\ $$$$\mathrm{u}\:\mathrm{may}\:\mathrm{be}\:\mathrm{1},\mathrm{3},\mathrm{7}\:\&\:\mathrm{9} \\ $$$$\mathrm{If}\:\mathrm{t}=\mathrm{u}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{are}\:\mathrm{11},\overset{×} {\mathrm{33}},\overset{×} {\mathrm{77}},\overset{×} {\mathrm{99}}\:\: \\ $$$$\mathrm{If}\:\mathrm{t}=\mathrm{10}−\mathrm{u}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{may}\:\mathrm{be} \\ $$$$\:\:\:\overset{×} {\mathrm{91}},\mathrm{73},\mathrm{37},\mathrm{19} \\ $$$$\mathrm{Total}\:\mathrm{such}\:\mathrm{numbers}\:\mathrm{which}\:\mathrm{have}\: \\ $$$$\mathrm{mentioned}\:\mathrm{property}: \\ $$$$\mathrm{11},\mathrm{33},\mathrm{77},\mathrm{99},\mathrm{91},\mathrm{73},\mathrm{37},\mathrm{19} \\ $$$$\mathrm{Among}\:\mathrm{which}\:\mathrm{primes}\:\mathrm{are}: \\ $$$$\mathrm{11},\mathrm{73},\mathrm{37},\mathrm{19}\:\:\left(\mathrm{Four}\right) \\ $$$$ \\ $$
Commented by Tinkutara last updated on 29/Jun/17
Thanks Sir!
$$\mathrm{Thanks}\:\mathrm{Sir}! \\ $$

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