When is a Number Divisible by Some Other Number?
A number is said to be divisible by another number if the quotient is a whole number and the remainder is 0. For example, if we divide 12 by 3 then,
The quotient is 4, which is a whole number and the remainder is 0. This means 12 is divisible by the number 3 and we can say that 3 is one of the factors of 12.
Factors and Factor Pairs:
Factors of a number are whole numbers which can divide another larger number whose quotient is a whole number, and the remainder is 0. For example, when 32 is divided by 4, the quotient is 8 and the remainder is 0. This implies that 32 is divisible by 4, hence, 4 is a factor of 32.
We can write whole numbers as the products of two factors. The two factors are called the factor pairs of the numbers. Let’s consider the earlier example. We know 4 multiplied by 8 is 32 implies 4 and 8 is a factor pair of 32.
Divisibility Rules of Numbers:
A number is always divisible by its factors. Some numbers have special divisibility rules that can be used to solve division based problems. We can do this by determining whether a number is divisible by a smaller whole number or not . These rules help us understand the factors and divisibility easily. The rules for some numbers are,
For example, if we want to check whether the number 345 is divisible by 2 and 3 or not.
So, we do not need to divide the number and check the divisibility. We can check the divisibility using the divisibility rules. The one’s digit of the number 345 is 5, which is odd, hence the number is an odd number. So it is not divisible by 2. Again,
The sum of the digits,
3 + 4 + 5 = 12
We know that 3 × 4 = 12. We also know that 12 is divisible by 3 and this means that the sum of the digits of the number 345 is divisible by 3. Therefore, as per the divisibility rule of 3, 345 is divisible by 3.
Similar procedure can be used to determine whether a given number is divisible by another number or not.
Solved Examples on Divisibility Rule:
Example 1: Find the factors and factor pairs of 36.
Solution: Using the divisibility rules and a bit of multiplication, we can find factors and make a factor table,
Therefore, the factors of 36 are 1,2,3, 4, and 6.
Again, from the factor table we also get the factor pairs as;
1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 and 6.
Example 2: Find the factors and factor pairs of 42.
Solution: Using the divisibility rules and some multiplication, we can find the factors and make a factor table,
Therefore, the factors of 42 are 1,2,3, and 6.
From the factor table we get the factor pairs as;
1 and 42, 2 and 21, 3 and 14 and 6 and 7.
Example 3: Check the divisibility of the number 425 by 2, 3, and 5.
Solution:
Divisor 2,
425 is an odd number. So, it is not divisible by 2.
Divisor 3,
The sum of the digits is 4 + 2 + 5 = 11. 11 is not divisible by 3. So, the number 425 is not divisible by 3.
Divisor 5,
The one’s digit of the number is 5. Therefore, it is divisible by 5.
The number 425 is not divisible by 2 and 3, but it is divisible by 5.
Example 4:
Three different men worked and produced 25, 23, and 27 chocolates and they were to be packed in 9 boxes of chocolates. Can the chocolates be packed equally in 9 boxes?
Solution: Before we can find out if we can pack them equally in 9 boxes, we need to find the total number of chocolates that have to be packed.
Total chocolates = 25 + 23 + 27 = 75 [Add]
So, now we should check if 75 chocolates can be packed in 9 boxes. To find that let us use the divisibility rule of the number 9.
The sum of the digits of the number 75 is 7 + 5 = 12. 12 is not divisible by 9.
Therefore, we cannot put the chocolates equally in the 9 boxes.
Example 5: The number below has 3 as a factor. What are the possible numbers?
4_- 9 or,
Solution:
As 3 is a factor of the number, the sum of the digits of the number must be divisible by 3.
The sum of the given two digits are 4 + 9 = 13
Now we need to look for multiples of 3 that are close to 13. These are, 15, 18, 21.
So, to make the sum 15 we can add, 15 – 13 = 2.
Again, to make the sum 18 we can add, 18 – 13 = 5.
Again, to make the sum 21 we can add, 21 – 13 = 8.
Therefore the possible digits are 2, 5 and 8.
So, the possible numbers are,
429, 459 and 489.
1 is the only number whose factor is 1 itself and no other number.
Prime numbers have only two factors, 1 and the number itself. For example, 7 has only two factors; 1 and 7.
If the one’s digit of any number is 0 then the number will be divisible by 10. For example, 60 is divisible by 10, as the unit digit of 60 is 0.
2 is a factor of 6. So, if any number is divisible by 6 then it will also be divisible by 2. So, 2 will be a factor of the number whose factor is 6.
For example,
18 = \(3\times2\) so, 6 is a factor of 18.
Again, 18 = \(3\times 3\times 2\) [put 6 = \(3\times2\) ] So, 2 is a factor of 18.
Therefore, 2 is always a factor of a number, whose factor is 6.