Real numbers are of two types: Decimal and Integers. The Integers have no decimal values like 0.33, 0.45; Decimals can be 0.23, 0.333... etc
Integers are of three types negative numbers like -1, -2 .. , zero and positive numbers like 1, 2, 3...
The negative numbers and zero are called non positive and zero and positive numbers are called non negative numbers.
Decimal numbers are of finite or terminating decimal types or infinite decimal types. The infinite decimal type is classified as rational numbers if they can be expressed in the form p / q or irrational number if they can't be expressed in the form p / q.
Properties:
HCF of 'x' and 'y' is G then HCF of x , (x+y) and x , (x-y) and (x+y) , (x-y) is also G.
Number Theory |
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Calculate the Sum of factors of a number:
Step 1: Get prime factors of a number say 240
240 = 2^{4} * 3^{1} * 5^{1}
Step 2: Sum of factors formula is
240 = (2^{0} + 2^{1} + 2^{2} + 2^{3} + 2^{4}) * (3^{0} + 3^{1}) * (5^{0} + 5^{1})
Step 3: 31*4*6 = 744
Calculate the Number of factors of a number:
Step 1: Get the prime factors of a number
240 = 2^{4} * 3^{1} * 5^{1}
Step 2: Number of factors of a number.
Number of factors = ( 4 + 1 ) * ( 1 + 1) * ( 1 + 1) = 5 * 2 * 2 = 20
Thus the powers of the numbers are increased by one and multiplied.
Calculate the sum and number of even factors of a number:
Step 1: Get the prime factors of a number
240 = 2^{4} * 3^{1} * 5^{1}
^{ }
Step 2: Sum of even factors
240 = 2^{4} * 3^{1} * 5^{1}
^{ }
Step 2: Sum of odd factors
30 = 2^{1} * 3^{1} * 5^{1}
^{ }
Step 2: Sum of factors formula is
30 = (2^{0} + 2^{1} ) * (3^{0} + 3^{1}) * (5^{0} + 5^{1})
Divisibility theory |
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Finding the number of zeroes in a factorial:
Step 1: Select a number and use the formula
Find zeroes in 127! = [127 / 5] + [127 / 5^{2}] + [127 / 5^{3}] + [127 / 5^{4}] ...
= 25 + 5 + 1 + 0 = 31
Remember that we ignore decimal values so after 3^{rd} equation remaining terms are all 0's.
Finding the highest power of a number in a factorial:
Step 1: Select a number and use the formula
Find highest power of 3 in 127! = [127 / 3] + [127 / 3^{2}] + [127 / 3^{3}] + [127 / 3^{4}] + [127 / 3^{5}] + ...
= 42 + 14 + 4 + 0 = 60
Remember that we ignore decimal values so after 4^{th} equation remaining terms are all 0's.
Finding the highest power of a composite number in a factorial:
Step 1: Select a number and use the formula
Find highest power of 15 in 127! but 15 is composite so prime factors are 3 * 5.
Step 2: Find highest power of each prime factor.
Find highest power of 3 in 127! = [127 / 3] + [127 / 3^{2}] + [127 / 3^{3}] + [127 / 3^{4}] + [127 / 3^{5}] + ...
= 42 + 14 + 4 + 0 = 60
Find highest power of 5 in 127! = [127 / 5] + [127 / 5^{2}] + [127 / 5^{3}] + [127 / 5^{4}] ...
= 25 + 5 + 1 + 0 = 31
Choose the lesser of both values and that is the answer = 31.
Factorial and Highest
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