__Factors that end with zero: __

Factors whose unit digit is zero or we can say factors which are divisible by 10.

**Example 1: Find the number of factors of 58800 that end with 0. **

**Solution:** If a number ends with 0 then it must have at least 2^{1} and 5^{1}.

We first factorize 58800.

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

Hence odd factor must have

2^{(1 or 2 or 3or 4)}—– 4 factors

3^{(0 or 1 ) } —– 1+1=2 factors

5^{(1 or 2)} ——- 2 factors

7^{(0 or 1 or 2) }— 1+2=3 factors

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

** Alternate way**:

To find number of factors, which are divisible by 10 we divide the given number by 10 and then find the number of factors of the quotient.

Divide 58800=2

^{4 }3

^{1}5

^{2}7

^{2}by 10 to get 2

^{3}3

^{1}5

^{1}7

^{2 }and its number of factors are 4x2x2x3=48

__Factors not ending with zero__

**Example 2**: **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 odd factors = Total number of factors – Number of factors ending with 0 =90- 48 =42

__Factors divisible by 12__

**Example 3**: **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}.

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) }— 1+2=3 factors

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

** Alternate way**:

To find number of factors, which are divisible by 12 we divide the given number by 12 and then find the number of factors of the quotient.

Divide 58800=2

^{4 }3

^{1}5

^{2}7

^{2}by 12 to get 2

^{2}5

^{2}7

^{2 }and its number of factors are 3x3x3=27