DIVISIBILITY RULES

DIVISIBILITY RULES
Divisibility Rule of 5
    Numbers, which last with digits, 0 or 5 are always divisible by 5
Example: 10, 10000, 10000005, 595, 396524850, etc
Properties Of Divisibility By 5:
    Last Digit Rule:
    The divisibility by 5 rule is straightforward. A number is divisible by 5 if its last digit is either 0 or 5.
    Repeated Patterns in Decimals:
    When expressing fractions with 5 in the denominator as decimals, the decimal part often exhibits repeating patterns.
    For example, 1/5 = 0.2, 2/5 = 0.4, 3/5 = 0.6, and so on. The digit 2 repeats in a cyclic pattern.
Addition and Subtraction Patterns:
    Alternating sums and differences of numbers in sequences that involve 5 often have interesting divisibility properties. For example, the alternating sum of consecutive odd numbers starting from 1 is always divisible by 5
Divisibility Rule of 6
    Numbers which are divisible by both 2 and 3 are divisible by 6. That is, if the last digit of the given number is even and the sum of its digits is a multiple of 3, then the given number is also a multiple of 6
Example: 630, the number is divisible by 2 as the last digit is 0.
    The sum of digits is 6+3+0 = 9, which is also divisible by 3.
    Hence, 630 is divisible by 6.
Properties Of Divisibility By 6:
    Divisibility by 2 and 3:
    A number is divisible by 6 if it is divisible by both 2 and 3. This is because 6 is the product of 2 and 3.
    Last Digit Rule for 2:
    For divisibility by 2, the last digit of the number must be even. Therefore, a number divisible by both 2 and 3 will have an even last digit.
    Sum of Digits Rule for 3:
    For divisibility by 3, the sum of the digits must be divisible by 3. Combining this with divisibility by 2, a number divisible by 6 will have an even sum of digits.
    Product of Consecutive Numbers:
    Any consecutive pair of numbers, one of which is even and the other is a multiple of 3, will result in a product divisible by 6. For example, 2 * 3 = 6, 4 * 3 = 12, and so on.
    Divisibility by 3 with the Alternating Sum Rule:
    Another rule for divisibility by 3 involves the alternating sum of the digits. If the alternating sum is divisible by 3, then the original number is divisible by 3. Combining this with divisibility by 2, a number divisible by 6 will satisfy both conditions.
Divisibility Rule of 7
    The rule for divisibility by 7 is a bit complicated which can be understood by the steps given below:

    Example: Is 1073 divisible by 7?
    From the rule stated remove 3 from the number and double it, which becomes 6.
    Remaining number becomes 107, so 107-6 = 101.
    Repeating the process one more time, we have 1 x 2 = 2.
    Remaining number 10 – 2 = 8.
    As 8 is not divisible by 7, hence the number 1073 is not divisible by 7.
Properties Of Divisibility By 7:
    Divisibility Patterns for Multiples:
    For multiples of 7, there are interesting patterns in the differences between consecutive multiples. For example, the differences between consecutive mul        tiples of 7 are 7, 14, 21, 28, and so on, forming a sequence of consecutive natural numbers.
    Divisibility by 7 and 11:
    When a number is divisible by both 7 and 11, its difference with the alternating sum of its digits is also divisible by 7.
    Casting Out 9s Method:
    An interesting method for checking divisibility by 7 involves "casting out 9s."Subtract twice the units digit from the rest of the number. If the result is divis ible by 7, then the original number is also divisible by 7.
    Benjamin Franklin's Square of 7 Rule:
    Benjamin Franklin discovered a rule for quickly determining if a number is a multiple of 7. If you square the units digit and subtract it from the rest of the         number, the result is divisible by 7.