Consider the following proposition. Let N be a two-digit number and let M be the number

Chapter 0, Problem 10

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Consider the following proposition. Let N be a two-digit number and let M be the number formed from N by reversing Ns digits. Now compare N2 and M2 . The digits of M2 are precisely those of N2 , but reversed. For example: 102 D 100 012 D 001 112 D 121 112 D 121 122 D 144 212 D 441 132 D 169 312 D 961 and so on. Here is a proof of the proposition: Proof. Since N is a two-digit number, we can write N D 10a C b where a and b are the digits of N. Since M is formed from N by reversing digits, M D 10b C a. Note that N2 D .10a C b/2 D 100a2 C 20ab C b 2 D .a2 / 100 C .2ab/ 10 C .b2 / 1, so the digits of N2 are, in order, a 2 ; 2ab; b2 . Likewise, M2 D .10b C a/2 D .b2 / 100 C .2ab/ 10 C .a2 / 1, so the digits of M2 are, in order, b 2 ; 2ab; a2 , exactly the reverse of N2 . Your job: Show that the proposition is false and explain why the proof is invalid. 11.