Prove a sum from divisibility of number theory.
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Answer:
Divisibility rules are efficient shortcut methods to check whether a given number is completely divisible by another number or not. These divisibility tests, though initially made only for the set of natural numbers (\mathbb N),(N), can be applied to the set of all integers (\mathbb Z)(Z) as well if we just ignore the signs and employ our divisibility rules. Note that the term "complete divisibility" means that one of the numbers with the smaller magnitude must be a divisor of the one with the greater magnitude.
But from where do these divisibility rules come from? How can we be so sure that it will work for every integer? Let us unveil the answers to all these questions as we explore the underlying principles behind this set of rules based on deductive reasoning and our knowledge of modular arithmetic.
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Divisibility Rules for some Selected Integers
Divisibility by 1: Every number is divisible by 11.
Divisibility by 2: The number should have 0, \ 2, \ 4, \ 6,0, 2, 4, 6, or 88 as the units digit.
Divisibility by 3: The sum of digits of the number must be divisible by 33.
Divisibility by 4: The number formed by the tens and units digit of the number must be divisible by 44.
Divisibility by 5: The number should have 00 or 55 as the units digit.
Divisibility by 6: The number should be divisible by both 22 and 33.
Divisibility by 7: The absolute difference between twice the units digit and the number formed by the rest of the digits must be divisible by 77 (this process can be repeated for many times until we arrive at a sufficiently small number).
Divisibility by 8: The number formed by the hundreds, tens and units digit of the number must be divisible by 88.
Divisibility by 9: The sum of digits of the number must be divisible by 99.
Divisibility by 10: The number should have 00 as the units digit.
Divisibility by 11: The absolute difference between the sum of alternate pairs of digits must be divisible by 1111.
Divisibility by 12: The number should be divisible by both 33 and 44.
Divisibility by 13: The sum of four times the units digits with the number formed by the rest of the digits must be divisible by 1313 (this process can be repeated for many times until we arrive at a sufficiently small number).
Divisibility by 25: The number formed by the tens and units digit of the number must be divisible by 2525.
Divisibility by 125: The number formed by the hundreds, tens and units digit of the number must be divisible by 125125.
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Answer:
Divisibility rules are efficient shortcut methods to check whether a given number is completely divisible by another number or not. These divisibility tests, though initially made only for the set of natural numbers (\mathbb N),(N), can be applied to the set of all integers (\mathbb Z)(Z) as well if we just ignore the signs and employ our divisibility rules. Note that the term "complete divisibility" means that one of the numbers with the smaller magnitude must be a divisor of the one with the greater magnitude.
But from where do these divisibility rules come from? How can we be so sure that it will work for every integer? Let us unveil the answers to all these questions as we explore the underlying principles behind this set of rules based on deductive reasoning and our knowledge of modular arithmetic.
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