The most common way to prove that an integer is divisible by another integer is by using the division algorithm. This algorithm states that if an integer a is divided by another integer b, the quotient q and remainder r can be expressed as a = bq + r, where 0 ≤ r < b. If the remainder is 0, then a is divisible by b.
An integer is divisible by 2 if the last digit is an even number. This is because all even numbers are divisible by 2, and the last digit of an integer represents the ones place. For example, the number 3542 is divisible by 2 because the last digit, 2, is an even number.
The rule for divisibility by 3 states that an integer is divisible by 3 if the sum of its digits is also divisible by 3. For example, the number 246 is divisible by 3 because 2 + 4 + 6 = 12, and 12 is divisible by 3.
An integer is divisible by 5 if the last digit is either 0 or 5. This is because all numbers that end in 0 or 5 are divisible by 5. For example, the number 12345 is divisible by 5 because the last digit is 5.
The divisibility rule for 7 is more complex, but one method is to use the fact that 7 × 11 = 77. This means that if the difference between an integer and 77 is divisible by 7, then the original integer is also divisible by 7. For example, the number 3087 is divisible by 7 because 3087 - 77 = 3010, and 3010 is divisible by 7.