Question

What is the max. Possible integer value of p if
7*8*9*... upto

135=n*720^p​

Answer 1

Given :- What is the max. Possible integer value of p if

7*8*9*_________ upto 135 = n*720^p

Solution :-

→ 7 * 8 * 9 _____ 135 = n * 720^p

since,

→ 720 = 2⁴ * 3² * 5

now,

→ (7 * 8 * 9 ______ 135) = 5^m

→ m = [(135/5) = 27 + (27/5) = 5] => 27 + 5 = 32 .

so, we can conclude that, in (7 * 8 * 9 ______ 135) maximum power of 5 is 32 .

similarly, in (7 * 8 * 9 ______ 135)

• maximum power of 9 :- 135/9 = 15 + (15/9) = 1 => 15 + 1 = 16 .
• then maximum power of 3² = 16 * 2 = 32 .

and, in (7 * 8 * 9 ______ 135)

• maximum power of 16 :- 135/16 = 8
• then maximum power of 2⁴ = 8 * 4 = 32 .

therefore, we can conclude that,

→ 7 * 8 * 9 _____ 135 = n * (2⁴ * 3² * 5)³²

Hence, the max. Possible integer value of p will be 32 .

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Answer 2

Given  : 7*8*9*... upto 135=n*720^p​

To Find :   max. Possible integer value of p  

Solution:

7*8*9*... upto 135=n*720^p​

720 = 2 * 2 * 2 * 2 * 3 * 3 * 5

=> 720 = 2⁴ * 3² * 5¹

7*8*9*... upto 135

Maximum power of 2 would be

[135/2] + [135/2²] + [135/2³]  + [135/2⁴] + [135/2⁵]  + [135/2⁶]   + [135/2⁷]  

- ( [6/2] + [6/2²]  )

2⁸ > 135         - is done as  as upto 6 products are missing

 [ ]   indicates greatest integer function

= 67 + 33 + 16 + 8  + 4 + 2  + 1  -  ( 3 + 1 )

= 131 - 4

= 127

as power should be 2^4p

[127/4] =   31    Hence  maximum value of p is  31  

Now for 3

[135/3] + [135/3²] + [135/3³]  + [135/3⁴] -  [6/3]  

= 45 + 15 + 5 + 1   - 2

= 64

as power should be 3^2p

Hence p = [64/2]  = 32

Now for 5

[135/5] + [135/5²] + [135/5³]  -  [6/5]

= 27 + 5 + 1 - 1

= 32

maximum value of p = 32

But least value in 31 , 32 and 32 is  31

Hence max. Possible integer value of p is  31

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