Number Properties Flashcards
When there’s only one pair of positive integers whose product equals a number, what can we infer about the number?
Y is a prime number.
Reason: If Y wasn’t prime, it would have divisors other than 1 and itself, leading to more than one pair of integers whose product equals Y.
Example 1: Y = 15
1. The pairs of positive integers whose product equals 15 are:
- 1 * 15 = 15
- 3 * 5 = 15
Here, we have two pairs of positive integers that multiply to give 15. This means 15 is not a prime number (which is true, as 15 is divisible by 1, 3, 5, and 15).
Example 2: Y = 7
1. The pairs of positive integers whose product equals 7 are:
- 1 * 7 = 7
In this case, there’s only one pair of positive integers that multiply to give 7. This suggests that 7 is a prime number (which is true, as 7 is only divisible by 1 and 7).
When forming a two-digit number from individual digits, how many possible orientations should be considered? By orientation I mean arrangments
Always consider both orientations.
Example: If digits are 3 and 1, the possible two-digit numbers are 31 and 13
When the product of numbers is even, what can we infer about the number?
At least one of the numbers is even. If all numbers were odd, their product would be odd.
What are the factors of a number x?
What are the factors of 12?
Factors of x are numbers that can divide x without leaving a remainder.
Example:
What are the factors of 12?
1,2,3,4,6 and 12.
Remember the table use, when we have to flip the number.
Which numbers have only 2 factors?
Prime numbers. They can divide by one and themselves (the prime number)
Which numbers have only 3 factos
Prime numbers squared.
What are the factors of 25?
1,5,25
When you add 1 to an addition of numbers (4+3), what can be said about the divisibility of the resulting number?
The resulting number is not divisible by any of the numbers in the original product.
For example, 6+1=7, and 7 is not divisible by 2 or 3, the factors of 6
When you add 1 to the product of x consecutive even numbers, what can be said about the smallest prime factor of the resulting number?
The smallest prime factor will be greater than X.
When Given Z = (10^n) - K and the sum of the digits of Z, how can you determine the structure of Z and solve for N?
1.- Recognize the structure: Z will have a series of nines followed by a number derived from the subtraction of K
2.- Borrowing Process: Subtracting K will cause borrowing, turning zeroes into nines until the initial 1 becomes 0.
3.- Equation from the sum: If X is the number of nines, the sum of the digits will be 9X plus the sum of the digits of the number derived from K.
4.- Solve for X: Use the given sum of the digits to solve for X.
5.- Determine N: N will be X plus the number of digits derived from K.
Example: For Z = (10^N) - 4 and sum of digits = 186, the equation is 9X + 6 = 186, leading to N = 21
Which is the multiple (number), that when divided by two will always give you an even result?
4
What results from raising any integer to any positive integer power, distinguishing between even and odd integers?
4^2 Even Integer
5^2 Odd Integer
- The result of raising any odd integer to any positive power is always odd
- The result of raising any even integer to any positive power is always even
What is the result of taking the square root of a perfect square, distinguishing between even and odd perfect squares?
- The square root of an even perfect square is always even
- The square root of an odd perfect square is always odd
When we see “not even/odd” in a question, what does it mean?
Odd/Even or fractions must be
When we see “non-negative / positive” in a question, what does it mean?
Positive / Negative or ZERO
What is the property of squaring real numbers?
a^2 ≥ 0
- The square of any real number is always non-negative, meaning positive
- A^2 is positive if a is non-zero
-A^2 is zero only if A is zero
