Happy NumbersDate: 01/05/2006 at 18:05:04 From: margie Subject: What is a "happy" number? What is a "happy" number? What are all the two-digit "happy" numbers between one and one hundred that are multiples of four? I have never heard the term. Date: 01/06/2006 at 00:55:39 From: Doctor Minter Subject: Re: What is a Hi Margie! A "happy number" is such that when you square each of its digits, add those squares, then repeat the process, eventually you will get an answer of 1. An example of a happy number is 7. Let's go through the process to see why 7 is considered "happy." 7^2 = 49 4^2 + 9^2 = 97 9^2 + 7^2 = 130 1^2 + 3^2 + 0^2 = 10 1^2 + 0^2 = 1 The fact that we ended up at 1 means that 7 is a happy number. Each number that we obtained along the way (49,97,130,10) are also happy numbers because we could have started the steps at any one of those numbers and gotten the same end result, as well as the same intermediate steps. The opposite is also true; that is, applying these steps to any number that is "unhappy" will only yield other unhappy numbers. The way to tell if a number is unhappy is when you keep applying the steps and find a repeated number. Once you find a repeated number, the process will repeat forever. Why these numbers are called "happy numbers" is beyond me. Maybe they ran out of terms, and someone was in a good mood when they thought this up. Not that I have any idea who "they" are. I digress... The commutative property of addition implies that any combination of digits of a (un)happy number will produce another (un)happy number. Therefore, since the above example showed that 49 and 97 are happy numbers, reversing their digits will show that 94 and 79 are happy as well. Also, since 64 is an unhappy number (try it!), 46 is unhappy as well. To find all of the happy numbers less than 100 that are multiples of 4, I advise first writing all multiples of 4 that are less than 100, and then evaluate and imply happiness or unhappiness of the others by the properties that I have mentioned. I mentioned that 64 is unhappy, so 8 is also, since applying one step to the number 8 will give a result of 64. I hope that this is enough to get you started on the specific problem. Please feel free to write again if you need further assistance or if you have any other questions. Thanks for using Dr. Math! - Doctor Minter, The Math Forum http://mathforum.org/dr.math/ |
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