the largest positive integer k such that 12^k divides (109)! is
Answers
Answer:
52
Step-by-step explanation:
the largest positive integer k such that 12^k divides (109)! is
109! = 109 * 108 * 107............................................3*2*1
12^k = ( 2 * 2 * 3)^k = 2^2k . 3^k
Number from 109 to 1 having 3 as factor
108 * 105 * 102 * .....................................* 9 * 6 * 3
= 3³⁶ ( 36 * 35 * 34 *...............................*3 * 2 * 1)
Taking only having 3 as factors
= 3³⁶ ( 36 * 33 * 30...................................* 9 * 6 *3)
= 3³⁶ * 3¹² ( 12 * 11 * 10 ........................* 3 * 2 * 1)
Taking further only having 3 as factors
= 3⁴⁸ * (12 * 9 * 6 *3)
= 3⁴⁸ * 3⁴ (4 * 3 * 2 *1)
= 3⁵² * 3
= 3⁵³
K = 53
Lets check for 2 also similarly
Number from 109 to 1 having 2 as factor
108 * 106 * 104 * .....................................* 6 * 4 * 2
= 2⁵⁴ ( 54 * 53 * 52 *...............................*3 * 2 * 1)
Taking only having 2 as factors
= 2⁵⁴ ( 54 * 52 * 50...................................* 6 * 4 *2)
=2⁵⁴ * 2²⁷ ( 27 * 26 * 25 ........................* 3 * 2 * 1)
Taking further only having 2 as factors
= 2⁸¹ * (26 * 24 * 22 *....................... * 6 * 4 * 2)
= 2⁸¹ * 2¹³ (13 * 12 * 11 *....................... * 3 * 2 * 1)
Taking further only having 2 as factors
= 2⁹⁴ * (12 * 10 * 8 * 6 * 4 *2)
= 2⁹⁴ * 2⁶ (6 * 5 * 4 * 3 * 2 *1)
Taking further only having 2 as factors
= 2¹⁰⁰ ( 6 * 4 * 2)
= 2¹⁰⁰ * 2³ (3 * 2 * 1)
Taking further only having 2 as factors
= 2¹⁰³ * 2
=2¹⁰⁴
= 4⁵²
2k = 104
k = 52
109! = 4⁵² * 3⁵³ * x = 3x * 12⁵²
so this can be divided by 12⁵²
Hence 52 is the largest positive integer such that 12^k divides (109)! is