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Try this beautiful problem from Integer based on Integers and Divisors useful for ISI B.Stat Entrance.

The number of positive integers which divide 240 (where both 1 and 240 are considered as divisors) is

- 20
- 18
- 16
- 12

Integer

Divisor

Number theory

But try the problem first...

Answer: 20

Source

Suggested Reading

TOMATO, Problem 98

Challenges and Thrills in Pre College Mathematics

First hint

We have to find out the number of positive integers which divide 240.so at first we have to find out the factors of 240...

\(240=2 \times 120\)

\(240=3 \times 80\)

\(240=4 \times 60\)

\(240=5 \times 48\)

\(240=6 \times 40\)

\(240=8 \times 30\)

\(240=10 \times 24\)

\(240=12 \times 20\)

\(240=15 \times 16\)

\(240=20 \times 12\)

\(240=24 \times 10\) ..............

so we notice that the divisors are repeat........

Can you now finish the problem ..........

Second Hint

We notice that after \(240=15 \times 16\) this stape all the factors are repeats.....so we have to calculate up to \(240=15 \times 16\) step only....

can you finish the problem........

Final Step

Therefore the total number of positive integers are \(1,2,3,4,5,6,8,10,12,15,20,24,30,40,48,60,80,120,240\) i.e \(20\)

- https://www.cheenta.com/problem-based-on-lcm-amc-8-2016-problem-20/
- https://www.youtube.com/watch?v=V01neV8qmh4

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