Quant Boosters  Kamal Lohia  Set 6

Here the expression has odd number of factors.
Which means it is a perfect square.Now if N is even expression is odd(e+e+o =o)
And if n is odd expression is still odd(o+o+o=o)
Thus, the square is of an odd positive number.Putting expression equal to 11^2 satisfies it
And n comes out to be 9
As n(n+3)=108But ,how to ensure that there is no other solution besides this for higher squares????


the gcd cant be greater than the difference of the two consecutive terms of the sequence
Diff= 2n+1
Equating to 41 gives n =20 , nd the numbers are 410 and 451...whose gcd is 41..59 on equating doesnt give two numbers which hv gcd as 59
Hence 41.

X=[10^20000/10^100+2]
X= 10^200002^200/ 10^100+2
X=2^200(A1)/2(B+1)
X=2^199(some expression)Hence highest power is 199
Is my solution right?

Remainder of (P3)!/p=(p1)/2
Remainder of (p4)!×(p3)/p =(p1)/2
Rem(p4)!/p = (1p)/6Hence rem( 6*(p4)!/p)= rem((1p)/p)= 1

12100 =2^2 ×5^2 ×11^2
110 =2×5×11Number of factors of 12100 less than 110 =
3×3×31/2= 13
Numbr of factors of 110 less than 110 =7Hence total factors of 12100 which are less than 110 but do not divide it are 137 =6

What is the logic to be used for tgis prob? Could anyone guide?

It will satisfy for 11th digit x=0,7

Could anyone suggest how to estimate that the series like
10^2 > 19^2 gives 1, 2, 3
20^2 > 29^2 gives 4, 5, 6, 7, 83 + 5 + 7 + 11 ....... which gives 100.
So It continues like this but fails for higher squares...which gives answer as 76 ...because the series of integers generated is not continuous.
So what is the right way to solve it?

In such scenarios like finding distinct values of [x^2/n] where x can be from 1, 2, 3 ... n
[1^2/n], [2^2/n] ... [(n/2)^2/n] will yield all numbers from 0 to [n/4] (means [n/4] + 1 distinct integers)
Then the next set (from [(n/2 + 1)^2/n] till [n^2/n] will be all different integers (means [n/2] distinct integers)
So the number of distinct integers would be [n/2] + [n/4] + 1if n = 100,
number of distinct integers would be [100/2] + [100/4] + 1 = 76if n = 2014,
number of distinct integers would be [2014/2] + [2014/4] + 1 = 1511if n = 13
number of distinct integers would be [13/2] + [13/4] + 1 = 10Just trying to generalize a solution shared by Kamal sir (Quant Boosters  Set 1  Q2).
You can try out with various numbers (may be smaller numbers) so that this can be verified.