Consider three natural numbers m, n and p such that the highest power of m contained in n! is p. Q1. If p=32 when m=2, then find p when m=3 Q2. What is the maximum number of distinct values of n that are possible for a given m and p, where the largest prime factor of m is a? Q3. If p =31 and m can be written as the product of two numbers in at least two ways, find the minimum possible value of n? Q4. If m can be written as the product of two distinct numbers in at least three ways, then find the maximum possible value of p when n = 36 |
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Consider three natural numbers m, n and p such that the highest power of m contained in n! is p.
Q1. If p=32 when m=2, then find p when m=3
a. 15
b. 17
c. 16
d. None of these
Soln. As p=32,m=2 thn n=34! or 35!.....
[34/3] + [34/3^2] + [34/3^3]=11+3+1=15....
For 35 nswer remains same.....
OPTION (a) is correct.....
Giv ur best to d world.
Nd d best wil cm back to u....
Regards,
Dipanjan.........
n/a
Q2. What is the maximum number of distinct values of n that are possible for a given m and p, where the largest prime factor of m is a?
a. a – 1
b. a
c. a + 1
d. m – a
Soln. As d no. of possibilities of 'n' depends on the [n/a] i.e d first power of a......
Dn d no of Max distinct values wud be a.....(Check it by takin xamples)
So option (b) is correct.....
Giv ur best to d world.
Nd d best wil cm back to u.....
Regards,
Dipanjan.......
n/a
Q3. If p =31 and m can be written as the product of two numbers in at least two ways, find the minimum possible value of n?
a. 61
b. 63
c. 64
d. 66
Soln If we take an xample dn it wud b clear.....
Bt d introspection is:
The min possible value is 2p+1 = 2x31+2 =64....
So option (c) is correct.....
Giv ur best to d world.
Nd d best will cm back to u.....
Regards,
Dipanjan......
n/a
HI Dipanjan.... Not able to understand ur solution.....
what i feel .... it should be D)..... i hav taken example n= 6;1*6 and 2*3 or take example of n= 12 ; 2*6 and 4 * 3.
th comman prime factor is 3.
Though i cant undrstand regardin which qs u r facin problm....Bt i m tryin to xplain qs no. 2
Q2. What is the maximum number of distinct values of n that are possible for a given m and p, where the largest prime factor of m is a?
a. a – 1
b. a
c. a + 1
d. m – a
Soln. Let m=6=2*3
D largest prime factor is a=3.
Let n=9,10,11.
for 9! p=[9/3] + [9/3^2]=4
for 10! p=[10/3]+[10/3^2]=4
for 11! p=[11/3]+[11/3^2]=4
Bt for 12! p=5 nd for 8! p=2.....
So max distinct values of n=3=a........
So i thnk option (b) is correct......
Giv ur best to d world.
Nd d best will cm back to u......
Regards,
Dipanjan......
n/a
I have always wanted a compendium of novena prayers. Thank you for sharing all these prayers with us. It brings joy and happiness to everyone. I know, I do feel that way.k
Q4. If m can be written as the product of two distinct numbers in at least three ways, then find the maximum possible value of p when n = 36
a. 12
b. 17
c. 18
d. 19
has any one solved it ????
123456789101112131415..............is ano. findd 28383rd term
If n=36, then p=?We get this soln. From2p+2=n2p+2=362p=34p=17so, option 2 will be d answer,
Am i right if not then could u tell me?
n/a
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