# Divisibility

**Divisibility**

Divisibility is an important concept in quantitative ability and mathematics in general. It deals with determining whether one number can be divided evenly by another without leaving a remainder. In other words, it checks whether one number is a multiple of another.

To understand divisibility, you need to be familiar with the following terms:

Dividend: The number that is being divided.

Divisor: The divisor is the number by which the dividend is being divided.

Quotient: The result of the division.

Remainder: It is the amount left over after the division.

When dealing with multiple numbers, you can apply these rules one by one to check the divisibility of each number. For example:

To check if 126 is divisible by 2, Since the last digit is 6 (even), it is divisible by 2.

To check if 354 is divisible by 3, The sum of its digits is 3 + 5 + 4 = 12, which is divisible by 3, so 354 is divisible by 3.

To check if 248 is divisible by 4, The last two digits are 48, which form a number divisible by 4, so 248 is divisible by 4.

To check if 520 is divisible by 5, The last digit is 0, which is divisible by 5, so 520 is divisible by 5.

Question 1:

Is the number 1,583,452 divisible by 2, 3, 5, or 7?

Solution 1:

To check divisibility by 2, we look at the last digit, which is 2 (even). So, 1,583,452 is divisible by 2.

To check divisibility by 3, we sum up the digits: 1 + 5 + 8 + 3 + 4 + 5 + 2 = 28. Since 28 is not divisible by 3, 1,583,452 is not divisible by 3.

To check divisibility by 5, we look at the last digit, which is 2 (not 0 or 5). So, 1,583,452 is not divisible by 5.

To check divisibility by 7, we can use long division or other methods. Long division shows that 1,583,452 ÷ 7 has a remainder, so 1,583,452 is not divisible by 7.

Question 2:

Find the smallest five-digit number that is divisible by both 6 and 9.

Solution 2:

To find the smallest five-digit number that is divisible by both 6 and 9, we need to find the least common multiple (LCM) of 6 and 9.

Prime factorization of 6: 2 × 3

Prime factorization of 9: 3 × 3

The LCM of 6 and 9 is the product of their highest powers: LCM(6, 9) = 2 × 3 × 3 = 18

The smallest five-digit number divisible by 18 is 18,000.

Question 3:

Is the number 52,601 divisible by 11?

Solution 3:

To check divisibility by 11, we take the alternating sum of the digits (starting from the left):

5 - 2 + 6 - 0 + 1 = 10

Since 10 is not divisible by 11, 52,601 is not divisible by 11.

Keep in mind that these divisibility rules are not applicable for all numbers, and there are more complex methods for divisibility testing for larger or more challenging numbers. However, the rules listed above are handy for everyday use and most common scenarios.