Veritas_Arithmetic

Odd Integer

Any integer that when divided by 2 does not result in a whole number.

Divisible by 2, if

If even

The ratio of kids aged 3, 4 and 5 at a kindergarten is 7:11:9, respectively. If the number of 3 year olds increases by 7, but number of 4 and 5 year olds stays the same, the new ratio will be?

It cannot be determined from the information given. We can't know how the addition of students will impact the ratio unless we know what the multiplier is and here we do not know what the multiplier is.

Multiples are infinite, factors are finite

Multiples are infinite, they go all the way up the number line and all the way down the number line.

Can a negative integer be prime?

No

Least Common Multiple

The smallest non-zero number that is a multiple of two or more numbers. (Key to figuring out LCMs is prime factorization)

If you add two numbers together and the result is positive, then

Then at least one of the numbers is positive.

How many unique factors does a number have?

(Using 222 as an example) 1. FIND all prime factors of a number. 2, 3, 37 2. EXPRESS the integer as a product of its prime factors. 2^1*3^1*37^1 3. STRIP off the exponents and discard the bases. So 1,1,1 4. ADD one to each exponent. So 2, 2, 2, 5. MULTIPLY those numbers together. So 2*2*2=8

1/9

.111

1/8

.125

1/7

.143

1/6

.1666

1/5

.2

Any number ending in a zero when multiplied to a power will end in what?

0

What are the unique factors of 72?

1 and 72, 2 and 36, 4 and 18, 8 and 9, 6 and 12, 3 and 24

Count to 50 only using prime factors.

1, 2, 3, 2^2, 5, 2*3, 7, 2^3, 3^2, 5*2, ect.

What is the tens digit of 11^13?

3. Find the pattern.

In a solution the ratio of alcohol to water by volume is 2.5. The perfect of alcohol by volume is closest to?

30% 2x + 5x = 7x 2/7= close to 30%

Non-negative integer

Could be zero or positive integer

Integer X represents the product of all integers between and 25, inclusive. The smallest prime factor of (x+1) must be: A) Between 1 and 10 B) Between 11 and 15 C) Between 15 and 20 D) Between 20 and 25 E) Greater than 25

E Adding one to the product breaks the number line that the original product was on therefore the greatest prime factor must be greater than any of the previous multiples.

Often it is easier to check multiples rather than factors when looking at divisibility.

For example: Is 209 a prime number? We know that 220 is divisible by 11, and 220-11 =209 therefore 209 is also divisible by 11 and is not prime.

Numbers that are not integers

Fractions, mixed numbers and decimals

is x>0? Data sufficiency 1) xy=16 2) x + y = 10

From statement one we learn that they're either both positive or both negative. From statement 2 we learn that at least one is positive. Taken together, we know that they both must be positive, and the answer is thus C, together the statements are sufficient. Here it is important to recognize that the unsung +/- property. If x and y sum to a positive number, they both can be negative.

Every product of two primes will have exactly four factors. For example, if a and b are both prime, then the factors of ab are 1, a, b, and ab.

If an exponent is applied, it won't give us new prime factors, it will tell us how many existing factors we'll have.

Divisible by 6, if

If even and sum of digits is divisible by 3

Divisible by 4, if

If last 2 digits divisible by 4

Divisible by 5, if

If last digit is 0 or 5

Divisible by 9, if

If sum of digits divisible by 9

Divisible by 3, if

If sum of digits is divisible by 3

Greatest Common Factor

The largest number that divides evenly into a set of numbers. (Key to figuring out GCFs is prime factorization)

Is 2 a prime number?

Yes. It is the only even prime number. It is also the lowest prime number.

What is zero a multiple of?

Zero is a multiple of everything, because zero x zero equals zero.

Some questions specifically ask about the units digit,

but even when it's not stated you can use this concept to scan answer choices saving shortcuts.

Even - odd =

odd

In N is divisible by an integer d, then

then it is a multiple of d.

Percent Change Calculation

(New Value - Original Value/Original Value) = Percent Change

What is the Greatest Common Factor of 110 and 66

22: 110 prime factors to 11, 2, 5, and 66 prime factors to 11, 2, 3. 2 and 11 are shared, so multiplying them together equals 22, which is the answer.

How many unique factors do 72, 100, 140 have?

72 has 12 unique factors. 100 has 9 unique factors. 140 has 12 unique factors.

How many multiples of 3 exist between -10 and 10

7: -9, -6, -3, 0, 3, 6, 9

The number of hours it took three drivers to drive a certain distance was in the ratio of 3:4:7. If they drove a combined 420 hours, how many hours did the trucker who drive the slowest drive?

210 3x + 4x + 7x = 420 14x = 420 x = 30 (If you consider that (3x + 4x) = 7x, then you realize that 7x is half of the overall hours. This can be seen as a shortcut)

Four supporting beams (north, east, south and west) are holding up a house with a weight distribution in the ratios 3:4:7:9. There is no other support for the house. If the eastern beam is holding 20,000 pounds, what is the total weight of the house?

115,000 4x= 20,000 Therefore, x=5,000 23(5,000) = 115,000 (Here x links the proportion to the total)

The ratio of kids aged 3, 4 and 5 at a kindergarten is 7:11:9, respectively. If the number of 3 year olds doubles, but number of 4 and 5 year olds stays the same, the new ratio will be?

14:11:9

What is the units digit of 15^9-16^6?

15 to any power ends in 5. 16 to any power ends in 6. Therefore XXXX5-XXX6 will end in 9.

At the peak of the real estate market, an investor bough a parcel of land. In the year that immediately followed, the value f that land decreased by 20%. By what percentage will the land need to appreciate for the investor to break even?

25% Decrease by 20% equals 4/5 of the original. How do you get 4/5(x) back to x? Multiply by 5/4. 80+x=100 x=20 20 is what percent of 80 20=x/100*80 20=8/10(x)x=25

What is the Least Common Multiple of 56 and 20

280: 2^3*7*5

Any number ending in a 5 when multiplied to a power will end in what?

5

How many unique factors does 20 have?

6: 1 and 20, 2 and 10, 4 and 5. Factors always come in pairs.

Divisor

A positive integer that divides an integer into another integer

What is a prime number?

A prime number is a positive integer with exactly two factors.

Three ways to express ratios

A:B The colloquial expression. This is how many problems will be written. A/B Expressing the proportion as a fraction. This is useful in calculation, but one must remember to keep track of units. ax + bx = total Equation form. This includes the multiplier (x) and the allows one to relate the proportion to the actual numbers involved, being mindful of the total amount.

Even Integer

Any integer that when divided by 2 results in a whole number, including zero.

Integer

Any whole number, including zero. Can be negative.

If x is an integer, is x/2 an even integer? Data sufficiency. 1. x is a multiple of 2 2. x is a multiple of 4

B. Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question asked. 2 is a multiple of 2, but 2/2 is 1 and one is odd. 4 is a multiple of 2, and 4/2 is even. Because both are possibilities, Statement one is not sufficient. 4 is a multiple of 4, and 4/2 is even. This is the same for all other multiples of 4 when divided by 2. Therefore statement 2 is sufficient. Another way to think about it using algebra: Is x/2 an even integer? Is x=2(an even integer) Is x=2(2(an integer) If x is a multiple of 4, then this must be true. Multiples of 4 are: 2(2)(even integer)

Is integer Y divisible by 24? Data sufficiency. 1) y is divisible by 6 2) y is divisible by 4

E: Statements 1 and 2 TOGETHER are not sufficient to answer the question and more information is needed. For example, both 12 and 24 are both divisible by both 6 and 4, but 12 is not divisible by 12. The fact that 12 could be an answer (there could be other answer such as 24 and 48), but it is not ruled out means that there is not enough information. Integer Y could be divisible by 24, but is not necessarily divisible by 24.

In order to give his customers a 25% discount on the price and still a net a 25% profit on the cost of an item, at what price should merchant mark an item if it cost him $16.80

E:$28

How high do you have to test for prime numbers?

If you take the number and look for the next highest perfect square, you only have to test up to the square root of that perfect square. For example, is 211 prime? 15^2=225, therefore test up to 15.

Multiple

Product of an integer and another integer. Can be negative.

If a number has an odd number of unique factors, than

Than it is a perfect square e.g. 36

When you encounter large numbers and multiplication,

Use Digit number properties to avoid tedious math.

Even - even =

even

Odd - odd =

even

Any number ending in a one when multiplied to a power will end in what?

one