# Given a positive integer N and two of its divisors. Difference between N and these two divisors is 270 and 280...

+1 vote
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We are given a positive integer N. Two of its positive divisors are chosen and the differences between N and these two divisors are 270 and 280 respectively.

Find the number of possible value(s) of N?

posted Jun 28

If 'a', 'b' are the 2 positive divisors of N & if (N - a) = 270 & (N - b) = 280, then a > b or in other words a = b + 10
a = N - 270 & b = N - 280, which means N has to start from 280 for 'a' & 'b' to be positive.

285(N) is divisible by 15(a) & 5(b) ====> 15*19 & 5*57 ====> LCM = 15 < 285
288(N) is divisible by 18(a) & 8(b) ====> 18*16 & 8*36 ====> LCM = 72 < 288
300(N) is divisible by 30(a) & 20(b) ====> 30*10 & 20*30 ====> LCM = 60 < 300
315(N) is divisible by 45(a) & 35(b) ====> 45*7 & 35*9 ====> LCM = 315 = 315
are 4 instances for which the above conditions held true. This is partly because as 'a' and 'b' becomes bigger compared to 'N', 'a' and 'b' stop having an LCM lesser than N. In other words 'a' & 'b' will not divide 'N' simultaneously.

Similar Puzzles

I'm thinking of a number.

• It is a two-digit positive integer.
• The sum of its digits is 10.
• Subtracting 72 from the number swaps its two digits.

What is the number?

See the following table

``````Number    Number of positive divisors
1           1
2*2         3
3*3*3       4
4*4*4*4     9
``````

As n increases, the number of positive divisors of nxnxnxn.....n (n times) increases. Is it true or false?

+1 vote

N has precisely 10 positive divisors.
N has precisely 15 positive divisors.
N has precisely 20 positive divisors.
N has precisely __ positive divisors.