divisibilty rules of 2 to 10

Answer: Divisibility rules from $2$ to $10$ : - $2$ : Last digit is even ( $0, 2, 4, 6, 8$ ). - $3$ : Sum of digits is divisible by $3$ . - $4$ : Number formed by last $2$ digits is divisible by $4$ . - $5$ : Last digit is $0$ or $5$ . - $6$ : Divisible by both $2$ and $3$ . - $7$ : Truncated number minus twice the last digit is divisible by $7$ (or $0$ ). - $8$ : Number formed by last $3$ digits is divisible by $8$ . - $9$ : Sum of digits is divisible by $9$ . - $10$ : Last digit is $0$ .

Step-by-step solution

Step 1: Divisibility by 2, 5, and 10 (Last Digit Tests)

A number is divisible by $2$ if its units digit is even ( $0$ , $2$ , $4$ , $6$ , or $8$ ). It is divisible by $5$ if its ones digit is either $0$ or $5$ . A number is divisible by $10$ if its units digit is strictly $0$ .

Step 2: Divisibility by 3 and 9 (Sum of Digits Tests)

A number is divisible by $3$ if the sum of all its digits is divisible by $3$ . Similarly, a number is divisible by $9$ if the sum of all its digits is a multiple of $9$ . For example, for $729$ , the sum of digits is $7 + 2 + 9 = 18$ , which is divisible by both $3$ and $9$ .

Step 3: Divisibility by 4 and 8 (Last Multiple Digits)

A number with $3$ or more digits is divisible by $4$ if the number formed by its last $2$ digits (tens and units) is divisible by $4$ . A number with $4$ or more digits is divisible by $8$ if the number formed by its last $3$ digits (hundreds, tens, and units) is divisible by $8$ .

Step 4: Divisibility by 6 (Composite Rule: 2 and 3)

Since $6 = 2 \times 3$ and $\gcd(2, 3) = 1$ , a number is divisible by $6$ if and only if it is divisible by both $2$ and $3$ . This means the number must be even and the sum of its digits must be a multiple of $3$ .

Step 5: Divisibility by 7 (Truncation and Doubling)

To test divisibility by $7$ , double the last digit and subtract it from the remaining truncated number. If this difference is $0$ or divisible by $7$ , then the original number is divisible by $7$ . For instance, for $343$ : $34 - (2 \times 3) = 34 - 6 = 28$ , which is divisible by $7$ .

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