Number System: Factors and Coprimes

Formula 6: The number of ways of writing a number N as a product of two co-prime numbers = 2n−1 where n=the number of prime factors of a number.

Example: Find the number of ways of writing 60 as a product of two co - primes
Ans: The prime factorization of 60 = 22×3×5
The number of ways of writing 60 as a product of two co - primes = 23−1 = 4

Formula 7: Product of all the factors of N = N⎛⎝Number of factors2⎞⎠ = N⎛⎝⎜(p+1).(q+1).(r+1)....2⎞⎠⎟

Example: Find the product of all the factors of 50
Ans: Prime factorization of 50 = 2×52 
Product of all the factors of 50 = 50(1+1).(2+1)2=503

                                                

1. P is the product of all the factors of 15552.  If P = 12N×M, where M is not a multiple of 12, then find the value of M.  [M and N are positive Integers]
Ans: 15552 = 26×35 
Product of all the factors of 15552 = (26×35)(6+1).(5+1)2=2126×3105 
⇒(22)63×363×342 = 1263×342 
So M = 342

2. Let M be the set of all the distinct factors of the number N=65×52×10,Which are perfect squares.  Find the product of the elements contained in the set M. 
N = 65×52×10 = 26×35×53
Even powers of 2 available: 20,22,24,26
Even powers of 3 available: 30,32,34
Even powers of 5 available: 50,52
Therefore number of factors of the number N that are perfect squares = 4 x 3 x 2 = 24
Product of the elements contained in M 
= 2(2+4+6)×6×3(2+4)×8×52×12=272×348×524


3. In a hostel there are 1000 students in 1000 rooms.  One day the hostel warden asked the student living in room 1 to close all the doors of the 1000 rooms.  Then he asked the person living in room 2 to go to the rooms which are multiples of his room number 2 and open them.  After he ordered the 3rd student to reverse the condition of the doors which are multiples of his room number 3.  If He ordered all the 1000 students like the same, Finally how many doors of those 1000 rooms are in open condition?
We understand that a door is in open or in close condition depends on how many people visited the room.
If a door is visited by odd number of persons it is in close condition, and is visited by even number of persons it is in open condition.
The number of people who visit a certain door is the number of factors of that number.  Let us say room no: 24 is visited by 1, 2, 3, 4, 6, 8, 12, 24 which are all factors of 24. Since the number of factors are even this door is in open condition.
we know that the factors of a number N=ap.bq.cr... are (p+1).(q+1).(r+1)...
From the above formula the product is even if any of p, q, r... are odd, but the product is odd when all of p, q, r are even numbers. 
If p, q, r ... are all even numbers then N=ap.bq.cr... is a perfect square.  
So for all the perfect squares below 1000 the doors are in closed condition.  
There are 31 perfect squares below 1000 so total doors which are in open condition are (1000-31)= 969 

4.  What is the product of all factors of the number N = 64×102 which are divisible by 5?
Sol: N = 64×102 = 26×34×52
Total product of the factors = (26×34×52)(6+1).(4+1).(2+1)2=(26×34×52)1052
So total product of the factors N which are not multiples of 5 = 
(26×34)(6+1).(4+1)2 = (26×34)352

So, total product of the factors of N which are multiples of 5 = (26×34×52)1052(26×34)352 = 2210×3140×5105

5.  Let N = 23×317×56×74 and M = 212×35×54×78.  P is total number of even factors of N such that they are not factors of M.  Q is the total number of even factors of M such that they are not factors of N.  Then 2P -Q = ?
Sol: 
N = 23×317×56×74 
M = 212×35×54×78

Calculation of P:
Number of powers of 2 available are 3. i.e., 21 to 23(all these powers are even)
Since P is the total number of even factors of N such that they are not factors of M, the number of powers of 3 available are 17 - 5 = 12
For combination with each of these 12 powers of 3( i.e., 36 to 317 , number of powers of 5 are 7 ( i.e., 50 to 56)  and number of powers of 7 are 5 ( i.e.,70 to 74) available respectively. 
Therefore there are 3 x 12 x 7 x 5 = 1260 factors

Now consider powers of 5 that are in N but not in M: 55,56
So, for these two powers of 5, the number of powers of 2, 3, and 7 that are available are 3, 6, and 5 respectively.  (Powers of 3 are only 6 (30 to 35), not 18 as we have already used 36 to 317 in caclulating P)
Therefore, there are 2 x 3 x 6 x 5 = 180 factors
P = 1260 + 180 = 1440

Calculation of Q:
Consider powers of 2 that are in M but not in N: 24,25,26.....212
So for these 9 powers of 2, number of even factors of M such that they are not factors of N = 9 x 6 x 5 x 9 = 2430
Consider powers of 7 that are in M but not in N: 75,76,77 and 78
So for these 4 powers of 7, number of powers of 2, 3, and 5 that are available are 3, 6 and 5. 
(Powers of 3 are only 6 (30 to 35), not 18 as we have already used 36 to 317 in calculating Q and (Powers of 5 are only 5 (50 to 54), not 7 as we have already used 55 to 56 in calculating Q))
So number of such factors = 4 x 3 x 6 x 5 = 360
Q = 2430 + 360 = 2790

Therefore, 2P - Q = 2880 - 2790 = 90