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## Question

At a math class, a teacher gives his students 4 numbers, lets say the numbers are A, B, C and D
Then the teacher ask his students to make an equallity equation from the numbers.

Claudy says  2*a*b = c + 2*b
Leon says    a*b*c = 12 * c
Jemmy says   b*b + c*c = (a+b)^2
Thomas says  30*d = 6*b*c
Richard says a+b+c+d = a*a

Then the teacher says "Ok students, you are smart, but one of you is wrong"

If the numbers are positive integer not more than 100, Who is wrong, and what are the numbers ?

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• 1

statement given by jemmy :  b*b + c*c = (a+b)^2  is incorrect.

initially lets assume that Leon's statement is correct :     a*b*c = 12 * c

That means  a * b = 12

which means there are following possible values of a  & b

a = 6  b = 2 or a = 2 b = 6

or

a = 4 b = 3 or a = 3 b = 4

now take a = 6 and b = 2 then

from   Claudy's statement   2*a*b = c + 2*b  => c = 20

and on using  Thomas's statement : 30*d = 6*b*c => d = 8

and  Richard's  statement :  a+b+c+d = a*a => 6 + 2 + 20 + 8 = 6 * 6 => 36 = 36

only  Jemmy's statement :  b*b + c*c = (a+b)^2 is not satisfying this values.

so jemmy's statement is incorrect.

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Spoiler

Beginning with the assumption that Leon's statement is correct, it can be found that Jemmy's statement must be incorrect while the others statements have solutions that can be correct. Yet, given the assumption that Leon's statement is incorrect, it can be found that at least one other student's statements must be incorrect. Without iterating through each possible deduction, I shall simply state that It can be mathematically deduced that Jemmy is incorrect and the values are:
A=6; B=2; C=20; and D=8.

To show that a solution does exist with the deduction that Claudia, Leon, Thomas, and Richard are correct:
Claudia
2AB = C + 2B
2(6)(2) = (20) + 2(2)
24 = 24

Leon
ABC = 12C
(6)(2)(20) = 12(20)
240 = 240

Thomas
30D = 6BC
30(8) = 6(2)(20)
240 = 240

Richard
A+B+C+D = A2
(6)+(2)+(20)+(8) = (6)2
36 = 36

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