Happy New Year! I start this new blog by posting some interesting facts about the number 2014. This post is mainly addressed to young students. 2014 is an even number. Let us see whether it is a perfect number. What is a perfect number?
If the sum of the positive proper divisors of a positive number equals the number, then it is called a perfect number. Here are some examples of perfect numbers
6=1+2+3
28=1+2+4+7+14
496=1+2+4+8+16+31+62+124+248
In the case of 2014,
2014 = 2X19X53.
So there are 8 factors for 2014. When you sum the proper factors of 2014, we get
1+2+19+53+38+106+1007 = 1226.
Since 1226 is less than 2014, 2014 is a deficient number.
The decimal system is the one that we use in day to day life. Roughly speaking, the digits 0,1,2,3,4,5,6,7,8,9 are used to express any number and perform arithmetic operations. For example, 123 has one hundred, two tens and three ones.
Rather, 123=1X10^2+2X10+3X1
Why should not we use the two digits 0 and 1 to express the numbers instead of ten digits?
Yes. We can. It is called the binary number system. The number 5 can be expressed as 101 in the binary form.
101 = 1X2^2+0X2+1X1=4+0+1=5.
If a number, expressed in binary form, contains odd number of 1s, then the number is called as an Odious number. On the other hand, if it contains even number of 1s, it is called as an Evil number.
The binary expansion of 2014 is 11111011110. , It has odd number of 1s. So 2014 is an Odious number
Even though 2014 is not a prime number, the sum of the digits and the sum of the cubes of the digits of 2014 are prime.
2+0+1+4=7 and 8+1+64=73.
Using the four symbols plus, minus, multiplication and division, 2010 can be expressed as
2010 = (7X 9+4)X 5X (6/2)X (3-1+8X 0).
Can you try expressing 2014 in a similar way?
That is, using the digits 1 2 3 4 5 6 7 8 9 0 once with the four basic arithmetic operations.
We will discuss about perfect numbers in detail in future posts.
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