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# Integer Properties: Practice Question of the Week #19

This week’s practice question looks deceivingly simple– try it out to test yourself before we post the answer tomorrow. Good luck!

2600 has how many positive divisors?

1. 6
2. 12
3. 18
4. 24
5. 48

### 7 Responses to Integer Properties: Practice Question of the Week #19

1. MN October 4, 2011 at 1:13 pm #

There are 24 divisors; these are,
1, 1 x 5, 1 x 13, 1 x 25, 1 x 5 x 13, 1 x 13 x 25,
(2^1), (2^1)x 5, (2^1)x 13, (2^1)x 25, (2^1)x 5 x 13, (2^1)x 13 x 25,
(2^2), (2^2)x 5, (2^2)x 13, (2^2)x 25, (2^2)x 5 x 13, (2^2)x 13 x 25,
(2^3), (2^3)x 5, (2^3)x 13, (2^3)x 25, (2^3)x 5 x 13, (2^3)x 13 x 25.
.
Method:
2600 = 2^3 x 5^2 x 13
.
Rule of thumb: Number of divisors = Product of (1 more than exponent of prime factors).
.
Interesting question:
What is the smallest number with 20 divisors? I think it’s 192. Do you agree?
.
MN

• Margarette October 4, 2011 at 2:58 pm #

Hi, MN

Your answer to the practice question is correct! As for your question, 192 has 14 positive divisors (2^6 * 3), and the smallest number with 20 divisors is 240, since its prime factorization is 2^4 * 3 * 5.

Best,
Margarette

• MN October 4, 2011 at 5:22 pm #

Thank you, Margarette.
.
And yes, it’s 240 – I was wrong earlier. Thanks again.

2. Hussain October 4, 2011 at 5:33 am #

answer is D , 24……. 2600 = 2^3*5^2*13^1
so divisors turned out to be 4*3*2=24

3. Vivek Roongta October 4, 2011 at 1:48 am #

2600 = 2^3 * 5^2 * 13

total positive divisors = (3+1)*(2+1)*(1+1)
= 24

4. Aashish October 3, 2011 at 10:36 pm #

the ans is D

5. t7t700 October 3, 2011 at 5:54 pm #