Can you find the number!

Logic Level 2

Let \otimes denote a binary operator such that for certain single digit positive integers A A and B B , the value of A B A \otimes B is a unique 3-digit positive integer.

We are given that 1 1 = 121 1 2 = 134 2 2 = 444 1 \otimes 1 = 121 \\ 1\otimes2 = 134 \\ 2 \otimes 2 = 444 What is the value of 2 3 2\otimes3 ?


The answer is 459.

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2 solutions

Mahdi Raza
Jun 19, 2020

Let \square denote an operation between a a and b b , which are two integers. Then, by the pattern, we can see: 1 1 = 1 2 ( 1 + 1 ) 1 2 1 2 = 1 2 ( 1 + 2 ) 2 2 a b = a 2 ( a + b ) b 2 \begin{aligned} \red{1} \square \blue{1} &= \overline{\red{1^2}(\red{1} + \blue{1})\blue{1^2}} \\ \red{1} \square \blue{2} &= \overline{\red{1^2}(\red{1} + \blue{2})\blue{2^2}} \\ & \ldots \\ \red{a} \square \blue{b} &= \overline{\red{a^2}(\red{a} + \blue{b})\blue{b^2}} \end{aligned}

Thus: 2 3 = 2 2 ( 2 + 3 ) 3 2 = 459 \begin{aligned} \red{2} \square \blue{3} &= \overline{\red{2^2}(\red{2} + \blue{3})\blue{3^2}} \\ &= \boxed{459} \end{aligned}

Shriniketan Ruppa
Jun 18, 2020

The digit in 100's place is the square of the first number in the equation. The digit in 10's place is equal to the sum of both the numbers. Unit digit is equal to the square of the second digit in the equation.So,

2 + 3=( 2 2 2^{2} )(2 + 3)( 3 2 3^{2} )

   =459

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