if 6k is a factor of 25! , what is the largest possible value of k
Answers
Step-by-step explanation:
Given :-
6k is a factor of 25!
To find :-
What is the largest possible value of k ?
Solution :-
Given number = 25!
If 6k is a factor of 25! then
6k divides 25! completely
=> 25! must be in the multiple of 6k
We know that
n! = n(n-1)(n-2)...1
=> 25! = 25×24×23!
=> 25! = 25×4×6×23!
=> 25! = 100×6×23!
=> 25! = 6×100×23!
=> 25! =6k
=> 6×100×23! = 6k
=> 6k = 6×100×23!
=> 6×k = 6×(100×23!)
On dividing by 6 both sides then
=> k = 100×23!
Therefore, k = 100×23!
If the given 6^k then
6^ k is a factor of 25! then
We have to find the number of 6's in 25!
So
6 = 2×3
So we have to find powers of 2 and 3 in 25!
Number of 2 's in 25!
=> 25/2 + 25/4 + 25/8 + 25/16
=> 12+6+3+1
=> 22
Number of powers of 2 = 22
Number of 3's
=> 25/3 + 25/9
=> 8+2
=> 10
Number of powers of 3's = 10
=> 25! = 2²²×3¹⁰
=> 25! = 2¹⁰×3¹⁰×2¹²
=> 25! = 6¹⁰× 2¹²
Since the powers of 2 > The powers of 3 then we have to calculate only powers of 3
6^k = 6¹⁰
K = 10
The possible value = 10
Answer:-
The largest possible value of k for the given problem is 10
Used formulae:-
- n! = n(n-1)(n-2)...1
Definition of n!:-
- The product of all positive integers less than or equal to n is called factorial n and it is denoted by n! .Where n is a non-negative integer .
Given :- If 6^k a factor of 25!, what is the largest possible value of k ?
a.6
b.4
c.9.
d.10
Solution :-
→ 25 ! = 6^k
→ 25 ! = (2 * 3)^k
→ 25 ! = 2^k * 3^k .
so, number of 2's in 25! ,
→ (25/2¹) + (25/2²) + (25/2³) + (25/2⁴)
→ (25/2) + (25/4) + (25/8) + (25/16)
→ 12 + 6 + 3 + 1
→ 22
and, number of 3's in 25!, { since 3³ > 25 .}
→ (25/3¹) + (25/3²)
→ (25/3) + (25/9)
→ 8 + 2
→ 10
then,
→ 25 ! = 2^22 * 3^10
→ 25 ! = 2^(10 + 12) * 3^10
→ 25 ! = 2^(10) * 2^(12) * 3^10
→ 25 ! = (2 * 3)^10 * 2^12
→ 25 ! = (6)^10 * 2^12 = 6^k .
therefore, the largest possible value of k will be (D) 10.
Learn more :-
let a and b positive integers such that 90 less than a+b less than 99 and 0.9 less than a/b less than 0.91. Find ab/46
p...
https://brainly.in/question/40043888