The power of 45 that will exactly divide 123! is? kindly post solution as well

1) 28
2) 30
3) 31
4) 59
5) 29

• 28
  45= 5*9=5*3^2
  123! has 123/5 +123/25 = 24+4=28 multiples of 5.
  also 123 has 123/3 +123/9 +127/27+127/81= 41+13+4+1=59 multiple of 3 or 29 multiples of 9.
  so 45^28 will exactly divide 123!
• 11 years agoHelpfull: Yes(1) No(0)
• 28
  45= 5*9=5*3^2
  123! has 123/5 +123/25 = 24+4=28 multiples of 5.
  also 123 has 123/3 +123/9 +127/27+127/81= 41+13+4+1=59 multiple of 3 or 29 multiples of 9.
  so 45^28 will exactly divide 123!
• 11 years agoHelpfull: Yes(0) No(0)
• Prime factorization of 45
  45 = 9•5
  45 = 3²•5
  
  
  So, we need to find the highest power of 3 and 5 that divides 123! exactly.
  
  The highest power of 3 that divides 123! is
  = ⌊123/3⌋ + ⌊123/3²⌋ + ⌊123/3³⌋ + ⌊123/3⁴⌋, where ⌊x⌋ denotes floor function of x
  = 41+13+4+1
  = 59
  
  The highest power of 5 that divides 123! is
  = ⌊123/5⌋ + ⌊123/5²⌋
  = 24+4
  = 28
  
  So 3⁵⁹ divides 123! exactly and 5²⁸ divides 123! exactly.
  
  45 is 9•5, we need 3⁵⁹ to change it into power of 9.
  3⁵⁹ = 3•(3²)²⁹
  3⁵⁹ = 3•9²⁹, the highes power of 9 that divides 123! is 29.
  
  Now, since the power of 5 is lesser than the power of 9. We take the highest power of 5.
  
  Therefore, the highest power of 45 that divides 123! is 28.
• 3 years agoHelpfull: Yes(0) No(0)