Concentration Improving Method for Exam

Distractions or Difficulties to concentration fall into two categories, internal or external.  Examples of internal Difficulties are hunger, drowsiness, daydreaming, and lack of interest, personal worries, and anxiety.

External Difficulties include environmental distractions such as other people, radio/television, outside noise, lack of proper study space.

The first step in improving concentration is to recognize the distraction and take action to correct it.

Internal Difficulties

Difficulties                                                                     Solutions

Hunger/drowsiness	Proper rest and nutrition are essential for good academic performance.  Try to get at least seven hours of sleep each night.  A diet rich in grains, vegetables, and fruit, but low in fat and sugar is recommended.  It will provide natural and sustained energy.
Daydreaming	If your mind starts to wander, jot down the interrupting thought so you can deal with it later.  In the meantime, continue to study.  Or, try the opposite approach, when  you start to drift off, recall the most important points of what you’ve just read, then allow yourself to daydream.  When you’re ready to return to your study materials, do so.
Lack of interest	Find a reason to be motivated.  Try to focus on positive reasons for doing well in the course such as higher GPA or personal growth.  Talk to your professor and classmates and examine their perspectives.
Anxiety/intimidation about studying	Learn the various skills and strategies that lead to effective studying.  If you are overwhelmed by the tasks at hand, break it down into smaller tasks.  Remember that planning and time management is crucial.
Personal worries	Identify the problem and take steps to resolve it.  Talk with someone who can help; a friend, a counselor or a specialist.

External Difficulties

Auditory – Conversation, radio, television, environmental noise	Avoid all conversation when studying.  Consider a “Do Not Disturb” sign for your dorm door to discourage visitors.  Turn off your cell phone.  While TV or radio can be distracting, continuous music (as found on a CD) might be conducive to studying.  Keep it low so that it is more like a background sound.  Instrumental music is probably preferable to music with lyrics.
Visual	Study in an area that lacks any visual stimuli that might distract you.  For example, if you study in the library, steer clear of the high traffic areas.  Stay away from windows or settings that cause your attention to be directed elsewhere. When studying, keep only the materials related to that subject in viewing range.  Don’t allow the temptation to be distracted by your other classes.
Disorganization	Before starting study sessions be sure to have all necessary material in front of you.  This eliminates the need to break concentration and search for an item.  Make a point of keeping all your study materials in a particular place and when it’s time to study you’ll know where everything is.

Always keep in mind that:

HIGHEST CONCENTRATION = MINIMAL DISTRACTION

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Number Theory Divisibility Methods

For checking divisibility by any 'prime/odd' number except for factors of 5', you have the concept of base number.

Number. Add Base Number Subtract Base Number

3: 1 2
7: 5 2
9: 1 8
11: 10 1
13: 4 9
17: 12 5
19: 2 17
21: 19 2
23: 7 ?
27: ? 8
29: 3 ?
...
...
And so on.... A method to get these described below 

Now for checking divisibility either add the last digit x add base number to the number "formed by removing last digit" or you can subtract last digit x subtract base number from the "formed by removing last digit"

e.g for check for 51/17 either 5- 1*5 =0 or 5 + 1*12 =17 hence divisible.
to check 312/13 we can 31+2*4 =39 hence dvisible or we can 31-2*9 = 13 ence divisible.
to check 61731/19 = 6173 + 1*2 = 6175 = 617 + 5*2 = 627 = 62 + 7*2 = 76 hence divisible.
to check 357976/29 = 35797 + 6*3 = 35815 = 3581 +5*3 = 3596 = 359 + 6*3 = 377 = 37 + 7*3 = 58 hence divisible..
to check 382294/11 = 38229-4*1 =38225 = 3822-5 = 3817 = 381 - 7 = 374 = 37 -4 =33 Hence divisible..


The SubractbaseNumber for a number can be obtained as the {(samllest multiple of number which ends in one)-1}/10
i.e. for 3 or 7 it is (21-1)/10 =2
for 13 it is 91-1/10 = 9.


The AddbaseNumber for a number can be obtained as the {(samllest multiple of number which ends in nine)+1}/10
i.e. for 13 it is (39+1)/10 =4.
for 7 it is 49+1/10 = 5

Proof:
For SubtactBaseNumber say the number abcde...
I want to check divisibility by 17 where subtractbasenumber is 5

I can always write abcde... as 10X+Y (where Y is last digit and X is number formed by removing last digit)

Now X-Y*5 = (10X -50Y)/10 = (10X + Y -51Y)/10 = (OriginalNumber - 51 Y ) / 10

The number '51 Y' is a multiple of 17 so if "OriginalNumber" is divisible by 17 then "OriginalNumber - 51*Y" got to be.. i.e. "10X - 50Y"
as 10 and 17 are co-prime if "10X- 50Y" is divisible the "X-5Y" got to be.....

same theory hold's for addbasenumbers too....


This also defines why it is so easy to check divisibility by 3 or 9 just keep on adding the digits...

And you can check divisibility by 11 just by keeping on subrating digits form previous number.. (which is same as taking sum of even/odd location separately..)

Divisibility by 7
Only for those interested in Number theory (Not a Cat short-cut)


say the number is :
38,391,787

Separate into pairs of digits
38 39 17 87

Consider the difference between each pair of digits and the nearest multiple of seven, beginning for the first pair at right, lower (upper) for the first, upper (lower) for the second and so on, alternating for each new pair.

4 -----4 (21-17)
38 39 17 87
---4 ------3 (87-84)

The resulting digits, read from right are 3444 (which is also a number multiple of 7).
Proceed in the same way with 3,444

1
34 44
----2

The final pair 21 is a multiple of seven, so is the original number 38,391,787.

ANOTHER EXAMPLE
Look how fast this method is.
Consider the 15-digit number 531,898,839,909,822
2 ----2--- 3 ----0
5 31 89 88 39 90 98 22
---3 ---4 ----6 ----1


Now we have 10,634,232
4 -----0
10 63 42 32
----0 ----4

And now 4,004

2
40 04
---4

Which gives 42, a multiple of 7.
We only need three steps for a 15-digit number.

This is called TOJA's method of divisibility. Incidentally this also works for 11 and 13. Just a little manupulation is required, (in case you get a remainder of more than 9)


Let A = 5,962
7
59 62 à 77 which is a multiple of 1
--7

EXAMPLE 2
Let A = 5, 971,845

6---- 4
5 97 18 45 à
--9 ----1

8
14 96 -> 88 ->divisible
----8


EXAMPLE 3
Let A = 80,714,546

8 ----10
80 71 45 46
----5-------2

The resulting numbers ( 2 10 5 8 ) don’t form a decimal number, so proceed in this way: Put the exceeding number 1 from 10, below the 2 and sum.

2 0 5 8 -> 3 0 5 8
1


3
30 58 à 33
---3




THE TOJA’S METHOD FOR DIVISIBILITY BY 13

EXAMPLE 1
Let A = 7, 046

8 (78 – 70)
70 46 à 78 which is a multiple of 13
---7 (46 – 39)



EXAMPLE 2

Let A = 9,581,681,629

----10 ----10
95 81 68 16 29
4-------3 ------3

--9
4 04 04
3 ----4

9
4 94 -> 39 divisible
--3 

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