Number Properties Flashcards
Leading Zeros
Leading zeros are all the zeros after the decimal point. i.e. 0.0005 has 3 leading zeros.
Rules:
1/x:
If x is an integer that is NOT a factor of 10, then substract 1 from the # of digits of x to obtain the # of leading zeros.
Ex. 1/124 has 2 leading zeros. 1/12 has 1 leading zero.
If x is an integer that is a factor of 10, then substract 1 from the # of trailing zeros or from exponent of x to obtain the # of leading zeros.
Ex 1/10 has 0 leading zeros. 1/100 has 1 leading zero. 1/1000 has 2 leading zeros. 1/10^20 has 19 leading zeros.
Trailing Zeros
Trailing Zeros are all the zeros after the last non-zero digit. Ex. 1,200 has 2 trailing zeros.
Trailing zeros are determined by the # of 2x5 pairs. Ex. 5^20 x 2^18 has 18 trailing zeros.
Number of Digits in a Number
To determine the # of digits in a number, factor the multiplication and find as many 2x5 pairs as you can. Each pair will add a trailing zero. Then multiply the remaining factors.
Ex. 50^8 x 8^3 x 11^2 = (552)^8 x (2^3)^3 x 11^2 = (2^11) x (5^16) x (11^2) = (2^17) + (5^16) + (11^2) = 2 x 10^16 x 11^2 = 2 x 121 x 10^16 = 242 x 10^16 –> 19 digits.
Division Properties of Factorials
The product of any set of consecutive integers is divisible by any of the integers in the set and any of the factor combinations of those numbers.
Ex: 5 x 6 x 7 x 8 = 1,680 which is divisible by 5, 6, 7, 8 and any combination of the factors such as 30, 35, 40, 42, 48, 56, 210
The product of any set of n consecutive integers (regardless if it starts at 1 or 5 or 235, etc.) will be divisible by n!
Ex: 5 x 6 is divisible by 2!
Ex: 3 x 4 x 5 x 6 is divisible by 4! (360 is divisible by 24)
Shortcut for Determining the Number of Primes in a Factorial
To determine the largest number of a prime number x that divides into y!, we perform the following steps:
Step 1:
Divide y by x, x^2, x^3, etc. Keep track of the quotients while ignoring remainders until you get to a quotient of zero.
Step 2:
Add the quotients from the division. The sum represents the number of prime number x in the prime factorization of y!
Step 2a:
If you need to determine the largest number of a non-prime integer that divides into y!, prime factorize the integer. Using the largest prime factor of x, apply the shortcut above to find the largest number of integer that divides into y!
Step 3a:
If you need to determine the largest number that is a power of a number that’s a factor of a factorial, prime factorize the integer. Then apply the shortcut above to find the largest number of the prime factor that divides into y!. Then divide that number by power of the number. It it doesn’t evenly divide, just use the quotient and ignore the remainder/decimal (round down).
Terminating Decimals
Terminating decimals have denominators that only have 2s, 5s, or both 2s and 5s as prime factors. If it includes anything else other than 2 or 5, it will NOT be terminating.
Remainder Theory
Remainders can be multiplied
Ex: What is the remainder of (12 x 13 x 17) / 5?
Find the remainders of 12/5, 13/5, and 17/5, and then multiple them together. So 2 x 3 x 2 = 12. Since 12 > 5, divide 12/5 to get a remainder of 2.
Remainders can be added & subtracted
Ex: What is the remainder of (12 + 13 + 17) / 5?
Find the remainders of 12/5, 13/5, and 17/5, and then add them together. So 2 + 3 + 2 = 7. Since 7 > 5, divide 7/5 to get a remainder of 2.
Ex: If you get a negative remainder with Absolute value less than divisor, then add the divisor to the negative remainder to get the answer.
Remainders have patters with prime factors raised to exponents.
Ex: x=3^(81n)/4 has a remainder pattern of 3(odd exponent) and 1 (even exponent). Remember that the remainder of x will vary unless n is specified to be even or odd, since it’s x raised to 81n. (think oddeven = even and oddodd = odd). If the expression was x=3^(80n/4), then the remainder would always be 1 because the exponent will always be even.
Range of Possible Remainders
The range of possible remainders of x/y is only determined by y. r max = y-1 and r min is always 0
Remainder Patterns
Remainder patterns resulting from dividends divided by the same integer will exist.
Ex. 0/5, r = 0
1/5, r = 1
2/5, r = 2
3/5, r = 3
4/5, r = 4
Remainder patterns resulting from integer powers divided by the same number
Ex. 2/7, r = 2
4/7, r = 4
8/7, r = 1
16/7, r = 2
32/7, r = 4
64/7, r = 1
Remainder when an integer is divided by 10^n is always the units digit of the integer.
Remainder when an integer is divided by 5, the remainder is always 5-UD if UD<5 and UD-5 if UD>5
Patterns in Units Digits
All powers of 0 end in 0 All powers of 1 end in 1 All powers of 2 end in 2, 4, 8, 6 All powers of 3 end in 3, 9, 7, 1 All powers of 4 end in 4, 6, 4, 6 All powers of 5 end in 5 All powers of 6 end in 6 All powers of 7 end in 7, 9, 3, 1 All powers of 8 end in 8, 4, 2, 6 All powers of 9 end in 9, 1
Consecutive Integers
Consecutive Integers will neve share the same prime factors. the GCF of two consecutive integers is always 1.
Evenly Spaced Sets
- A set of consecutive integers (included sets of even and odd consecutive integers)
- A set of consecutive multiples of a given number such as [2, 4, 6, 8], [10, 20, 30, 40] or [33, 66, 99, 132]
- A set of consecutive numbers with a given remainder when divided by some integer [1, 6, 11 ,16, 21] or [3, 7, 11, 15, 19]
Ex. a set of numbers with remainder 2 when divided by 6: [14, 20, 26, 32].
LCM & GCF Rule
If you know the LCM(x,y) and the GCF(x,y), then xy = LCM(x,y) * GCF(x,y)
Shortcut for Determining the # of Factors in an Integer
Prime factorize the integer. add 1 to all the exponents of the prime factors. multiply across.
How many factors does 520 have?
520 = (2^3)(5)(13)
4x2x2 = 4x4 = 16
There are 16 factors in 520.
Remainder Patterns to solve for divisibility
Ex: What number must be substracted from 2^256 so the result is a multiple of 3?
Use remainder pattern theory.
2^1/3 --> r = 2 2^2/3 --> r = 1 2^3/3 --> r = 2 2^even power divided by 3 --> r = 1 2^odd power divided by 3 --> r = 2
2^256 is even power, so r=1. Substract 1 from 2^256 to get a multiple of 3 (i.e. r=0)
