Question

sum of all divisors of 5400 whose units digit is 0

Answer 1

Solution:-
Sum of all the divisors whose units digit is 0.
Prime Factors of 5400 = 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5
= 2³ × 3³ × 5²
Sum of all the divisors of 5400 = (2⁰ + 2¹ + 2² + 2³) × (3⁰ + 3¹ + 3² + 3³) × (5⁰ + 5¹ + 5²)
 This will be the sum of all the divisors of 5400 but we have to find out the divisors of 5400 whose units digit is 0. It means that we have to find out all divisors of 5400, which are divisible by 10. So, we need to ensure that every term in the expression has 2 and 5 in it (as 2 multiplied by 5 = 10). Now, look at the above brackets of 2 and find the term that do not yield a 2 in the expression. Also check the bracket of 5 for the terms that do not yield 5. In these two brackets 2⁰ and 5⁰ are the culprits. We have to remove these two entries and we will get the expression that we need. This will be as follows.
Sum of all the divisors whose units digit are 0 :-
= (2¹ + 2² + 2³) × (3⁰ + 3¹ + 3² + 3³) × (5¹ + 5²)
= (2 + 4 + 8) × (1 + 3 + 9 + 27) × (5 + 25)
= 14 × 40 × 30
= 16800
So, the sum of all the divisors of 5400 whose units digit is 0 is 16800.
Answer.