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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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