Divisibility Rules: A Review

Recall from your grade school mathematics that the division algorithm for whole numbers has 4 basic components:

• the dividend - which is the number to be divided,
• the divisor - which is the number that will the divide the dividend,
• the quotient - which is the answer to a division problem, and
• the remainder - which is what remains after the division is executed

The concept of "divisibility" arises from the fact that every number can be divided evenly by a unique set of numbers. This means such a number can be divided with a remainder of zero. Consider, for instance, the number 1,456. Suppose it is divided by the numbers 2, 3, 4, 5, and 6. The quotients and remainders are as follows:

• 1,456 / 2 = 728 r 0
• 1,456 / 3 = 485 r 1
• 1,456 / 4 = 364 r 0
• 1,456 / 5 = 291 r 1
• 1,456 / 6 = 242 r 4

From the results, we can say that 1,456 is divisible by 2 and 4 because of the zero remainder, but not by 3, 5, and 6 because of the non-zero remainders.

The divisibility rules that follow enable us to see if a number is divisible by another number without actually dividing. Aside from these rules, having stock knowledge of your multiplication tables will come in handy.


Divisibility by 1 - All whole numbers are divisible by 1. Recall that any number divided by 1 results in that same number.

Divisibility by 2 - All even numbers are divisible by 2.

Divisibility by 3 - A number is divisible by 3 if the sum of its digits is a multiple of 3.

Divisibility by 4 - Look at the last 2 digits. If they are a multiple of 4 or are zeros, then the entire number is divisible by 4.

Divisibility by 5 - A number is divisible by 5 if the last digit is either 0 or 5.

Divisibility by 6 - An even number divisible by 3 is divisible by 6.

Divisibility by 7 -

• Take the last digit in a number.
• Double and subtract the last digit in your number from the rest of the digits.
• Repeat the process for larger numbers.
• Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7.

Divisibility by 8 - Look at the last 3 digits. If they are a multiple of 8 or are zeros, then the entire number is divisible by 8.

Divisibility by 9 - A number is divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility by 10 - All numbers that end in 0 are divisible by 10.

Divisibility by 11 -

• Starting with the left, take every second digit and find their sum.
• Take the sum of the remaining digits.
• Subtract the sum in step 2 from the sum in step 1.
• If the result in step 3 is zero or is divisible by 11, then the entire number is divisible by 11.

Divisibility by 12 - A number divisible by both 3 and 4 is divisible by 12.


Exercises:

• For each item, determine if the first number is divisible by the second number. 
• Then confirm your answer by doing the division.

1)   5,000; 8
2)  18,931; 11
3)   858; 4
4)  8,099; 7
5)  111,111,111; 9
6)  7,428; 12
7)  5,571; 3
8)  1,001; 7
9)  3,008; 8
10) 8,349; 11

7 ‘Good’ Habits That Are Bad For Your Health (from realbuzz.com)

Drinking water

While we are probably all familiar with the advice to drink eight glasses of water a day, more recent research has suggested that there is actually no scientific evidence supporting this recommendation and that drinking excessive amounts of water can actually be dangerous by lowering the concentration of salt in your blood. Health-conscious water drinkers should also be wary of the trend for drinking bottled water, as studies have suggested that the chemicals (phthalates) from plastic bottles can leach into water and disrupt hormone levels.

Talking over your problems

Talking through your problems can be a great way to gain some perspective and get things off your chest. However, studies have suggested that, after a certain point, rehashing and dwelling on problems can actually be bad for your health. According to research, revisiting and analyzing the same problems with friends (“co-rumination”) can lead to anxiety, stress disorders and depression. Next time a problem arises, by all means talk it over with a friend, but try to focus on problem-solving rather than simply dwelling on the issue.

Sipping on mocktails

You may think that by swapping cocktails for mocktails you are doing your health a favour, but this may not actually be the case. While cutting down on alcohol is beneficial for your wellbeing, mocktails are often high in refined sugar which research suggests is just as damaging and addictive as alcohol. For a safer swap and a shot of nutrients, make sure you stick to mocktails made from pure fruit juices instead of those made from syrups.

Early morning workouts

While a daily workout is great for your health, studies suggest that getting up for early morning exercise may not be as ideal as it seems. A study by a researcher from Brunel University, Middlesex, found that heavy training sessions early in the morning can compromise the immune system and put athletes at increased risk of bacterial and viral infection. While a morning jog or gentle exercise session is unlikely to put you at risk, it may be better to save heavier workouts for later in the day.

Taking nutritional supplements

We all know that vitamins are good for us, but relying on nutritional supplements can actually be bad for your health. Separate studies have shown that high doses of vitamin supplements including iron, magnesium and vitamin B6 raise the death rate of older women, while taking vitamin E can increase men’s risk of prostate cancer. While certain people may be required to take vitamins (those with low levels of vitamin D, for example, or vegans who may be deficient in vitamin B12), for most people a better approach is to opt for a varied diet full of fruit and vegetables which will give you all the nutrients you need.

Slathering on sunscreen

Official advice for many years has warned about the dangers of skin cancer, causing many of us to take measures to cover up in the sun at all times. However, while it is extremely important to protect your skin, experts have more recently advised that little and frequent sun exposure is good for us, preventing vitamin D deficiency, which can lead to rickets, osteomalacia and depression. Official advice in the UK, where rickets has recently made a comeback, is to spend 10 minutes in the midday sun without sunblock each day before covering skin up.

Switching to low fat foods

When getting started in healthy eating, it is tempting to opt for low fat foods in order to help keep off excess pounds. However, cutting out ‘good’ fats such as omega-3 fatty acids could be detrimental to your health. Omega-3 fatty acids, found in oily fish, walnuts and flaxseeds, not only help to keep skin supple and wrinkle-free, they are also essential for good brain and heart health and can help prevent arthritis.


Source: http://ph.she.yahoo.com/7-good-habits-bad-health-090000188.html

Senseless (by David Archuleta)

You're telling me it's over now
It's like you turned the volume down
I can't even hear myself.
Looks like you're about to leave
Now it's getting kinda dark to me
Cuz that aint something I can see.
It's like I'm getting worse everyday
Girl I'm breaking down.
Got my heart on the floor
And I can't feel it, 
Everything's numb.

Now I'm senseless
Got no feeling left.
My pain will let this heart beat in my chest,
Now I'm senseless.
Why are we doing this?
I'm so losing it, oh oh, senseless.

Feels like I lost your love
Since I had to give you up.
And now I'm so outta touch, yeah.
Another day, I start the same
It came across a punishment
A bitter taste is lingering.
My senses are tingling
Escaped, all the feeling's gone.
I'm cold and lost with no direction.
Everything's numb.

Now I'm senseless
Got no feeling left.
My pain won't let this heart beat in my chest,
Now I'm senseless.
Why are we doing this?
I'm so losing it, oh oh

Think it smells like rain water on the floor.
But I don't know cuz I'm so gone
Can't even see tomorrow coming on
Can't even hear myself scream,
Can't even feel anything.
Now I'd die to love you if I could.

Cuz girl I'm so senseless
Got no feelings left.
My pain will let this heart beat in my chest,
Senseless.
Why are we doing this?
I'm so losing it, oh oh.

Oh, senseless.
Oh, senseless.

The Alchemist (by Paulo Coelho)

My first ever video since I learned how to use Movie Maker. Enjoy watching.

Net 25 Interview with PCS President Dr. Lemuel Brana

PCS President Dr. Lemuel Brana has expressed support to the Automated elections in the Philippines. Watch the video for the details.

Redundancy in Speech

It is common knowledge to us that there are words that are sounded and are spelled differently but mean the same thing. These are what we call synonyms. There are also words that substitute for a noun if the latter is to be used many times in a passage. These are called pronouns.

The concept of redundancy is a common "illness" of many people whenever they communicate with other people. It is the manner of indicating an idea in a needless repetitive fashion. Unfortunately, there are a lot of people who think they are good speakers or writers but still appear to be "trying hard" because of redundancy.

Here are some instances of how we commit redundancy, followed by some tips:

1. Using the same noun over and over again. Hearing the same noun over and over again not only is annoying to the ear; it also wastes space in paper. To avoid these, pronouns are useful. See the examples below:

- "The mule has a built-in protection from being some animal's dinner: it tastes bad." ("it" for "mule")
- "Minnie is very vocal about her feelings for Joey. Unfortunately, he disses her because of her stalking attitude."
     ("her" for "Minnie" and "he" for "Joey")

2. Using the same adjective or verb over and over again. Describing nouns or actions using the same adjectives or verbs can be very monotonous and painfully boring for a reader, expert or amateur. It is here that synonyms (words of the same meaning) and antonyms(words of the opposite meaning) are useful. See the examples below:

- "An ideal match is for an intelligent man to be married to an equally educated woman."
      ("educated" is a synonym for "intelligent")
- Like David Cook and Eminem who are southpaw performers, David Archuleta is a left-handed singer."
      ("left-handed" is a synonym for "southpaw")
- The dot product is commutative, while the cross product is anti-commutative.
      ("anti-commutative" is an antonym for "commutative")

3. Using "double negatives." Negating a negation when in fact you really intend to convey a negative statement can be very confusing. Just as the product of two negative integers is a positive number, negating a negative sentence must indicate an affirmative one. So if you mean the opposite of something, negate only once. Here are some examples:

- "Some of your words are misspelled incorrectly." ("misspelled" could be changed to "spelled")
- "There won't be no next time" ("won't" can be replaced with "will" or "no" could be changed to "a")

4. Stating the explicit together with the implied. Implying something and then saying it in a separate word is redundant. It can be so funny it reveals one's weakness in employing common sense in speech. It can also be annoying to your audience, since no one likes to be treated as some dumbsh*t that needs much explanation. So, if you imply something, leave it as it is, unless it needs further elaboration. Some examples follow:

- "Please make me some coffee. I'll drink it.
     (Coffee is a drink. If you asked for it, it's obvious you want something to drink.)
- "John and Dean are two male brothers."
     (Brothers are male, so there's no need to say "male brothers", just "brothers.")
- "Come, let's eat some pizza pie."
     (Pizza means pie, so if you mean pizza, just refer to it as pizza, but if you mean some other pie,
      say "apple pie", then refer to it as pie.)

Operations on Decimals (A Review)

For years we have been operating on numbers that are based on powers of 10. These are called decimal numbers. For example, 231 is a decimal number because it is composed of  (2 x 100) + (3 x 10) + (1x1).

Meanwhile, there are decimal numbers that have components less than 1. To accommodate such numbers, we denote them with decimal points, and their place values are read similarly with whole number places but with "th" at the end and the decimal point is read as "and". For example, 231.045 is read as "two hundred thirty-one and forty-five thousandths".

Just as there are four basic arithmetic operations with whole numbers, there are also four basic operations with decimals. We review them here.

Addition and Subtraction
1. Align the decimal points.
2. Add normally.

Multiplication
1. Multiply normally.
2. Count the number of decimal places in both factors. Their total is the number of decimal places in the product.

Division
1. If there are decimal places in the divisor, move its decimal point to the right as many decimal places as possible to get rid of any decimal parts. For instance, if there are 3 decimal places, move the decimal point three places to the right.
2. Move the decimal point in the dividend in the same number of places as in the divisor.
3. Divide normally.


Examples.
1) 0.00056 + 0.32443 = 0.32499
2) 65.43 - 38.14 = 27.29
3) 0.3 x 0.6 = 0.18  (one decimal place in 0.3 and one in 0.6, so there are two decimal places in 0.18)
4) 24 / 0.12 = 2400 / 12 = 200 ( there are 2 decimal places in 0.12, so both dividend and divisor had their decimal points moved two decimal places)