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A two-digit number has the property that the square of its tens digit plus ten times its units digit equals the square of its units digit plus ten times its tens digit. Determine all two-digit numbers which have this property, and are prime numbers.

 Nov 26, 2019
 #1
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+1

91

19

73

37

55

I think these are the only numbers

 Nov 26, 2019
 #2
avatar+105989 
+1

A two-digit number has the property that the square of its tens digit plus ten times its units digit equals the square of its units digit plus ten times its tens digit. Determine all two-digit numbers which have this property, and are prime numbers.

 

let the number be   10a+b  where a and b are single digit integers and a is not 0

 

the square of its tens digit  (a^2)  plus ten times its units digit (10b) equals the square of its units digit (b^2) plus ten times its tens digit (10a)

 

\(a^2+10b=b^2+10a\\ a^2-10a=b^2-10b\)

 

If you cannot see the the answer off hand then try looking at a= 1 up to a =9  one by one and see if you can find a b value that will work.

 Nov 26, 2019
 #4
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0

a=1;b=0;p=0; cycle:d=a*10+b;if(a^2+10*b==b^2+10*a, goto loop, goto next); loop:printd," ",;p=p+1; next:b++;if(b<10, goto cycle, 0);b=0;a++;if(a<10, goto cycle, 0);print"Total = ",p

 

11  19   37    73   = 4 such numbers

 Nov 26, 2019
 #5
avatar+105989 
0

lets look at 19

 

1^2+90=91

9^2+10=91

 

Yes that does work :)

Melody  Nov 26, 2019

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