NUMBER+FUN+FOR+THE+SUMMER

AMAZING NUMBERS This first set of activities is about the special characteristics of some regular everyday numbers. Even the simplest numbers have some very special properties. We would like you to discover some of them.

**12** You already know a lot about the number 12.
 * There are 12 months in a year, there are 12 signs of the Zodiac, and there are 12 hours, repeated through each day and night.
 * 12 is divisible by the sum of its digits, 12 / 3 = 4, and by their product, 12 / 2 = 6.

Now you are going to learn something new about 12. There are only 21 abundant numbers not greater than 100, starting 12, 18, 20 … Can you find the other ones?
 * 12 is the first **ABUNDANT NUMBER**, meaning that it is less than the sum of its factors excluding itself: 1 + 2 + 3 + 4 + 6 = 16 and 12 < 16.
 * 8 on the other hand is an example of a **DEFICIENT NUMBER**, meaning that it is greater than the sum of its factors excluding itself: 1 + 2 + 4 = 7 and 8 > 7

12 < 16 || JAMES || 18 < 21 || MAR || 20 < 22 || ANE || 21 > 11 || ANE || 24 We would like you to list the first 15 numbers that are also divisible by the sum and the product of their digits.
 * = ==NUMBER ..... == || ==SUM OF FACTORS ... == || ==ABUNDANT ...... == || ==NAME ..... == ||
 * = 12 || 1 + 2+ 3 +4 + 6 = 16 || Yes, because
 * = 18 || 1 + 2 + 3 + 6 + 9 = 21 || Yes, because
 * = 20 || 1 + 2 + 4 + 5 + 10 = 22 || Yes, because
 * = 21 || 1 + 3 + 7 = 11 || No, because
 * The number of hours in a day.
 * Some mathematicians define the proper divisors of a number as all the divisors except for 1 and the number itself. If we use this definition then 24 is very special number because: 24 is the smallest composite number, the product of whose proper divisors is a cube:2 x 3 x 4 x 6 x 8 x 12 = 243.
 * 24 is also divisible by the sum of its digits, 24 / (2 + 4) = 4, and by their product, 24 / (2 x 4) = 3.

Now for another activity about the number 24. Choose four diferent numbers from 1 to 6 and use the basic operations to get 24.
 * = ==NUMBER ... == || ==SUM OF DIGITS .... == || ==PRODUCT OF DIGITS .... == || == .. NAME . == ||
 * = 12 || 3 and 12 / 3 = 4 || 2 and 12 / 2 = 6 || ANE ||
 * = 24 || 6 and 24 / 6 = 4 || 8 and 24 / 8 = 3 || GEMMA ||
 * = 12 || 3 and 12 / 3 = 4 || 2 and 12 / 2 = 6 || ANE ||
 * = 24 || 6 and 24 / 6 = 4 || 8 and 24 / 8 = 3 || GEMMA ||
 * = 12 || 3 and 12 / 3 = 4 || 2 and 12 / 2 = 6 || ANE ||
 * = 24 || 6 and 24 / 6 = 4 || 8 and 24 / 8 = 3 || GEMMA ||
 * = 12 || 3 and 12 / 3 = 4 || 2 and 12 / 2 = 6 || ANE ||
 * = 24 || 6 and 24 / 6 = 4 || 8 and 24 / 8 = 3 || GEMMA ||
 * = 12 || 3 and 12 / 3 = 4 || 2 and 12 / 2 = 6 || ANE ||
 * = 24 || 6 and 24 / 6 = 4 || 8 and 24 / 8 = 3 || GEMMA ||
 * = 24 || 6 and 24 / 6 = 4 || 8 and 24 / 8 = 3 || GEMMA ||

There are 15 diferent ways to choose the four numbers and it is possible to get 24 for all of the possible numbers. <span style="color: #ff9d00; display: block; font-family: Verdana,Geneva,sans-serif; font-size: 120%; text-align: center;">28 <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">28 is special because it is neither **//abundant//** nor **//deficient//**. It is what we call a **perfect number.** This is because the sum of its factors including 1but excluding itself is 28! 1 + 2 + 4 + 7 + 14 = 28.
 * = ==NUMBERS CHOSEN ......... == ||= OPERATION == == THAT RESULTS IN 24 == == ||= == ..... NAME .. . .. == ||
 * = 2, 3, 4, 6 || 2 x 4 x (6 – 3) = 24 || CECILIA ||
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">The number of days in February except in leap years.
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">It corresponds to the number of days in the lunar cycle.

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">28 is the second perfect number. The first four perfect numbers were known to the Greeks. Can you list the first four perfect numbers?

<span style="color: #ff9d00; display: block; font-family: Verdana,Geneva,sans-serif; font-size: 120%; text-align: center;">100 <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">There is an old puzzle that involves joining the digits 1 to 9 in that order, using only the usual signs of operations, and brackets, to make a total of 100. There are many solutions to the puzzle, here are two: 123 – 45 – 67 + 89 = 100, or in reverse order, 98 – 76 +54 + 3 +21 = 100.
 * ==List of first four perfect numbers ...kkkkkkk.......... == || == ............ NAME== ||
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">100 is the square of 10, the base of the decimal system.
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">It is also the boiling point of water on the Celsius scale of temperature.
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">Denoted by C by the Romans, from centum meaning hundred.
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">In the metric system, the prefix “centi” means one-hundredth, as in centimeter, one-hundredth of a metre.

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">Can you think of any others? Remember, you can use brackets. <span style="color: #ff9d00; display: block; font-family: Verdana,Geneva,sans-serif; font-size: 120%; text-align: center;">113 <span style="color: #ff9d00; display: block; font-family: Verdana,Geneva,sans-serif; font-size: 120%; text-align: center;">200 <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">We would like you to show how all the 3-digit composite numbers smaller than 200 can be changed into prime numbers by changing only one digit. Add your composite number to the list and then show how to convert it to a prime number. <span style="color: #ff9d00; display: block; font-family: Verdana,Geneva,sans-serif; font-size: 120%; text-align: center;">1024 <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">We would like you to express the number 1024 as a product of three integers in as many ways as possible. You can use the TAB key to add more rows to the table if necessary.
 * = ==SOLUTION hhhhhhhhhhhhhhhhhhhh.. == ||= == .kkkkkkkkkkkkkkk NAME== ||
 * 123 – 45 – 67 + 89 = 100 ||= JAMES ||
 * 98 – 76 + 54 + 3 +21 = 100 ||= CECILIA ||
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">This is the smallest 3-digit prime number such that all other arrangements of its digits are also prime numbers.That means 131 and 311 are also prime numbers. 199 and 337 also have this property. We would like you to investigate how many 2-digit prime numbers also have this property. There are only five.
 * = ==PRIME NUMBER . .. == || == .... OTHER ARRANGEMENTS .... == || == .......... NAME== ||
 * = 13 || 31 || ANE ||
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">This is the smallest number which cannot be changed into a prime number by changing 1 digit. Changing the hundreds digit gives 000, 100, 300, 400, 500, 600, 700, 800, 900, none of which are prime. Nor are 210, 220, 230 … Nor 201, 202, 203 …
 * = ==COMPOSITE NUMBER .... == ||= == .... CONVERTED TO PRIME ... == ||= == .... NAME== ||
 * = 100 ||= 101 ||= JAMES ||
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">1024 = 2<span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%; vertical-align: super;">10 and therefore the smallest number with 10 prime factors.
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">Although kilo- in the metric system usually means a thousand, as in kilogramme, 1K of memory in a computer means 1024 bytes of memory.
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">It is a nice coincidence that 2<span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%; vertical-align: super;">10 is so close to 10<span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%; vertical-align: super;">3
 * = ==NUMBERS AA..AA == ||= == aa.....aa PRODUCT A..........A == ||= == A......A NAME== ||
 * = 2, 4, 128 ||= 2 X 4 X 128 = 1024 ||= MAR ||

These activities are based on the book THE PENGUIN DICTIONARY OF CURIOUS AND INTERESTING NUMBERS by DAVID WELLS.