Factors | Algebra - Number Theory
What is the number of factors of 151200?
Answer: 144
Explanation :
For a number n = ax × by × cz
Number of factors = (x +1)(y + 1)(z + 1)
151200 = 25 × 33 × 52 × 7
∴ The number of factors of 70560 = (5 + 1)(3 + 1)(2 + 1)(1 + 1) = 6 × 4 × 3 × 2 = 144.
Hence, 144.
Workspace:
Find the sum of the factors of 360.
Answer: 1170
Explanation :
For a number n = ax × by × cz
Sum of factors = (ax + ax-1 + ... + 1)(by + by-1 + ... + 1)(cz + cz-1 + ... + 1)
360 = 23 × 32 × 5
Sum of factors of 360 = (23 + 22 + 21 + 1)(32 + 31 + 1)(51 + 1)
∴ The sum of the factors of 360 = = 1170
Hence, 1170.
Workspace:
How many factors of 14400 are odd?
Answer: 9
Explanation :
14400 = 26 × 32 × 52
For factor to be odd, the power of 2 should be 0.
∴ Possible power of 2 in factor = 0, i.e., only 1 possibility
There is no restriction of powers of other prime factors.
⇒ The number of odd factors of 14400 = (1)(2 + 1)(2 + 1) = 9
Note: The odd factors of 14400 are 1, 3, 5, 9, 15, 25, 45, 75, 225.
Hence, 9.
Workspace:
How many factors of 14400 are even?
Answer: 54
Explanation :
14400 = 26 × 32 × 52
For factor to be even, the least power of 2 should be 1.
∴ Possible power of 2 in factor = 1 or 2 … or 6, i.e., 6 possibilities
There is no restriction of powers of other prime factors.
⇒ The number of even factors of 14400 = (6)(2 + 1)(2 + 1) = 54
Hence, 54.
Workspace:
In how many ways can 1925 be written as a product of two co-primes?
Answer: 4
Explanation :
For a number n = ax × by × cz
Number of ways of writing n as a product of two co-prime factors = 2m-1, where m is the number of discting prime factor of n.
1925 = 52 × 7 × 11
∴ The number of ways in which 1925 can be expressed as a product of two co-primes is 23 – 1 = 4 (∵ there are 3 distinct prime factors i.e., 5, 7 and 11).
Note: The four ways in which 91091 can be expressed as a product of two co-primes is
25 × 77
11 × 175
7 × 275
1925 × 1
Hence, 4.
Workspace:
How many numbers lesser than 36 are co-prime to it?
Answer: 12
Explanation :
For a number n = ax × by × cz
Number of co-primes of n, less than n is =
36 = 22 × 32
∴ The number of co-primes of 72, less than 72 is:
= 12
Note: The co-primes of 36, less than 36 are 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35.
Hence, 12.
Workspace:
Find the sum of co-primes of 36 which are less than 36.
Answer: 216
Explanation :
For a number n = ax × by × cz
Sum of co-primes of n, less than n is =
36 = 22 × 32
∴ The sum of co-primes less than 36 = = 216.
Note: 1 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 25 + 29 + 31 + 35 = 216.
Hence, 216.
Workspace:
Find the total number of factors of 104 divisible by 26.
Answer: 3
Explanation :
104 = 23 × 13
26 = 2 × 13.
For factor to be divisible by 26, the least power of 2 should be 1 and that of 13 also should be 1.
∴ Possible power of 2 in factor = 1 or 2 or 3, i.e. 3 possibilities
Similarly, possible power of 13 in factor = 1, i.e. 1 possibility
⇒ The total number of factors of 104 divisible by 26 = (3) × (1) = 3.
Hence, 3.
Workspace:
How many factors of 5625 are not perfect squares?
Answer: 9
Explanation :
5625 = 32 × 54
For factor to be a perfect square, the power of prime factors should be even.
Let us first calculare number of factors which can be perfect squares
∴ Possible powers of 3 in factor = 0 or 2 i.e., 2 possibilities
Similarly, possible powers of 5 in factor = 0 or 2 or 4 i.e., 3 possibilities
⇒ Number of factors which are perfect squares = 2 × 3 = 6 factors.
Also, total factors of 5625 = (2 + 1)(4 + 1) = 3 × 5 = 15 factors.
∴ Number of factors which are not perfect squares = 15 – 6 = 9.
Hence, 9.
Workspace:
Find the total number of sets of factors of 264 which are co-prime to each other.
Answer: 31
Explanation :
For a number n = ax × by × cz
The number of sets of factors that are co-prime to each other = (x + 1)(y + 1)(z + 1) - 1 + xy + yz + xz + 3xyz
⇒ 264 = 23 × 31 × 111
∴ x = 3; y = 1; z =1
∴ The number of sets of factors that are co-prime to each other = (3 + 1)(1 + 1)(1 + 1) – 1 + 3 + 1 + 3 + 3 (3 × 1 × 1) = (4 × 2 × 2) + 6 + 9 = 31
Hence, 31.
Workspace:
In how many ways can 360 be written as product of two of its factors?
Answer: 12
Explanation :
Number of ways of writing a non-perfect square as a product of two of its factors is equal to half the number of its factors.
360 = 23 × 32 × 5
Number of factors of 360 = (3 + 1)(2 + 1)(1 + 1) = 24
∴ Number of ways of writing 360 as a product of two of its factors = ½ × 24 = 12 ways.
[1×360, 2×180, 3×120, 4×90, 5×72, 6×60, 8×45, 9×40, 10×36, 12×30, 15×24, 18×20]
Hence, 12.
Workspace:
In how many ways can 144 be written as product of two of its distinct factors?
Answer: 7
Explanation :
Number of ways of writing a perfect square as a product of two of its distinct factors is equal to half the (number of its factors - 1). Here 144 is a perfect square.
144 = 24 × 32
Number of factors of 144 = (4 + 1)(2 + 1) = 15
Number of ways of writing a perfect square as a product of two of its distinct factors = ½ × (15 - 1) = 7
[1×144, 2×72, 3×48, 4×36, 6×24, 8×18, 9×16]
Hence, 7.
Workspace:
In how many ways can 144 be written as product of two of its factors?
Answer: 8
Explanation :
Number of ways of writing a perfect square as a product of two of its factors (same or distinct) is equal to half the (number of its factors + 1). Here 144 is a perfect square.
144 = 24 × 32
Number of factors of 144 = (4 + 1)(2 + 1) = 15
Number of ways of writing a perfect square as a product of two of its distinct factors = ½ × (15 + 1) = 8
[Note: 12 × 12 is one additional way of writing 144 as product of two of its same factors]
Hence, 8.
Workspace:
How many factors of 344 are divisible by 4?
Answer: 4
Explanation :
344 = 23 × 43
For a factor to be divisible by 4 (= 22), the minimum power of 2 in it should be 2.
∴ Possible powers of 2 in the factor = 2 or 3, i.e., 2 possibilities.
Their is no restriction on powers of other prime numbers
∴ Possible powers of 43 = 0 or 1, i.e., 2 possibilities.
Total number of such factors = 2 × 2 = 4.
Hence, 4.
Workspace:
What is the product of all the factors of 360?
- (a)
36012
- (b)
36024
- (c)
36011
- (d)
None of these
Answer: Option A
Explanation :
360 = 23 × 32 × 5
Product of all the factors of N = N(number of factors)/2
Number of factors of 360 = (3 + 1)(2 + 1)(1 + 1) = 24
∴ Product of all factors of 360 = 36024/2 = 36012
Hence, option (a).
Workspace:
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