Understanding Divisors of 105
To find how many divisors of 105 have at least one zero at their end, we first need to analyze the number 105.
Prime Factorization of 105
- The prime factorization of 105 is:
105 = 3 × 5 × 7
Divisor Properties
- A divisor of a number has a zero at its end if it is a multiple of 10.
- Since 10 = 2 × 5, a divisor must include both the factors 2 and 5.
Factors of 105
- From the prime factorization, we see that 105 does not contain the factor 2.
- Therefore, none of the divisors of 105 can include 2.
Conclusion on Divisors with Zero
- Since 105 has no factor of 2, it cannot form any multiples of 10.
- Hence, none of the divisors of 105 will have at least one zero at its end.
Counting Divisors of 105
- To ensure completeness, let’s count the total number of divisors of 105:
- The formula for the number of divisors based on the prime factorization (p1^e1 × p2^e2 × ... × pk^ek) is:
(e1 + 1) × (e2 + 1) × ... × (ek + 1)
- For 105, the exponents are all 1 (e1 = 1 for 3, e2 = 1 for 5, e3 = 1 for 7):
(1 + 1) × (1 + 1) × (1 + 1) = 2 × 2 × 2 = 8.
Final Statement
- There are 8 divisors of 105, and none of them can have at least one zero at their end.
- Therefore, the answer to the question regarding divisors of 105 with at least one zero is incorrect; the correct answer is 0, not 12, 15, or 25.
This analysis shows that the answer provided (option 'D') is actually misleading based on the properties of divisors.