In this article we shall study more about factors. Hence, we’ll learn how to calculate the number of factors ending with zero, factors not ending with zero and factors divisible by various numbers(for eg: 12).

**Factors that end with zero:**

Factors whose unit digit is zero are the factors that are divisible by 10.

** ****Example**: Find the number of factors of 58800 that end with 0.

**Solution:**

We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

We basically need to find the factors that are divisible by 10. If a number ends with 0 then it must have at least one pair of 2^{1} and 5^{1}.

Here, we won’t consider

2^{(1 or 2 or 3or 4)}—– 4 factors

3^{(0 or 1 ) } —– 2 factors

5^{(1 or 2)} ——- 2 factors

7^{(0 or 1 or 2) }— 3 factors

Hence, total number of factors ending with 0 = (4)(2)(2)(3) = 48

**Factors not ending with zero**

**Example**: Find the number of factors of 58800 that are not ending with 0.

**Solution:** We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

** **Total number of factors of 58800 is 5x2x3x3=90 and in previous example we calculated total number of factors ending with 0 are 48.

Number of factors not ending with 0 = Total number of factors – Number of factors ending with 0

=90- 48 =42.

**Factors divisible by 12:**

**Example**: Find the number of factors of 58800, which are divisible by 12.

**Solution:**

Since we have to find the number of factors which are divisible by 12, then it must

have at least 2^{2} and 3^{1}. So we will not consider

We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

Hence factors divisible by 12 must have

2^{(2 or 3or 4)}—– 3 factors

3^{( 1 ) } —– 1 factor

5^{(0 or1 or 2)} ——- 3 factors

7^{(0 or 1 or 2) }— 3 factors

Hence, total number of factors, which are divisible by 12 = (3)(1)(3)(3) = 27

**NOTE: **To find the number of factors that are divisible by any composite factors, the maximum power in divisor must be present in the dividend.

** **To be thorough with these types of questions, solve the given exercise.

**EXERCISE:**

**Question 1:** Find the number of factors of 58800, which are divisible by 98.

(1) 12

(2) 10

(3) 6

(4) 18

### Answer and Explanation

Solution: Option 1

We first factorize 58800.

58800 = 2^{4 }3^{1}5^{2}7^{2}

Since we have to find the number of factors, which are divisible by 98 , then it must

have at least 7^{2} and 2^{1}. So we will not consider

Hence factors divisible by 98 must have

2^{(1 or 2 or 3or 4)}—– 4 factors

3^{( 0 or 1 ) } —– 2 factors

5^{(0 or1 or 2)} ——- 3 factors

7^{( 2) }———– =1 factor

Hence, total number of factors, which are divisible by 98 = > (4)(1)(3)(1) = 12

**Question 2:** Find the number of factors of 4200, which are divisible by 125.

(1) 5

(2) 1

(3) 2

(4) 0

### Answer and Explanation

**Solution: Option 4**

We first factorize 4200 = 2^{3 }3^{1}5^{2}7^{1}

Since we have to find the number of factors, which are divisible by 125 , then it must

have at least 5^{3 }.

But we can see that there is no 5^{3} . So no factor is divisible by 125.

**Question 3.** Find the number of factors of 600 that end with 0.

(1) 8

(2) 12

(3) 16

(4) 24

### Answer and Explanation

**Solution: Option 2**

Prime factorization of 600 i.e. 600= 2^{3}3^{1}5^{2}

We basically need to find the factors which are divisible by 10 . If a number ends with 0 then it must have atleast one pair of 2^{1} and 5^{1}.

So we won’t consider 2^{0} and 5^{0}. Now calculate number of factors by making combinations.

2^{(1 or 2 or 3)}—– 3 factors

3^{(0 or 1 ) } —– 1+1=2 factors

5^{(1 or 2)} ——- 2 factors

Hence, total number of factors ending with 0 = (3)(2)(2) = 12

**Question 4.** Find the number of factors of 600 that do not end with 0.

(1) 8

(2) 12

(3) 16

(4) 24

### Answer and Explanation

**Solution: Option 2**

** **Solution:

Prime factorization of 600 i.e. 600= 2^{3}3^{1}5^{2}

Total number of factors = (3+1)*(1+1)*(2+1)=24

Total number of factors ending with 0 = (3)(2)(2) = 12 { as we calculated in previous question }

Number of factors not ending with 0 = Total number of factors – Number of factors ending with 0

Number of factors not ending with 0= 24-12= 12