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According to the answer it is $3(2+1)2$ . This is because there are $3$ ways of dealing with the number $2$ (we can’t reject 2) similarly we can’t reject the number $5$ so there are $2$ ways of dealing with $5$ but there are $2+1$ ways of dealing with the number $3$ since we can accept it or reject it. However , doesn’t this also count a case when all the numbers are accepted , resulting in a factor that is the number itself ? Then why don’t we subtract one from the final result (since we’re looking for ‘proper’ divisors) ?

Aditi
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2 Answers2

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$1800 = 2^2*3^2*5*10$

Since we need that the divisor is divisible by 10, we chose this way.

Now for all the factors above, we have choice of chosing from them (except for 10).

So answer is 3*3*2=18

Proper divisors = 18-1=17

Sagar Chand
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We can write $1800 = 2^33^25^2$. Now for a number to be divisible by $10$, atleast one $2$ and one $5$ must appear. So we have can choose

  • One from $2,2^2, 2^3$
  • One from $3^0, 3^1, 3^2$
  • One from $5^1, 5^2$

So in all, we have $3\times 3\times2 = 18$ factors.

Now as you say, the number itself is included ($1800$) in this. If you need proper factors, then they are $18-1 = \boxed{17}$ in number.

jonsno
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