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Question 7 : If N and x are positive integers such that NN = 2160 and N2 + 2N is an integral multiple of 2x, then the largest possible x is [TITA]


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Explanatory Answer

Method of solving this CAT Question from Number Theory

Given that N and X are positive integers such that NN = 2160 and N2 + 2N is an integral multiple of 2x
Now we have to find the largest possible value for x
NN = 2160
We can rewrite this 2160 as (2p)q
such that 2p = N = q
pq = 160

We can try by substituting p and q with 2 and 80 respectively
(22)80 = 480
This doesn’t work so we can try by substituting p and q with 4 and 40 respectively
(24)40 = 1640
This also doesn’t work so now we can substitute p and q with 5 and 32 respectively
Such that (25)32 = 3232

Now we can say that N = 32
N2 + 2N ⟹ 322 + 232 = 2x × k (where k is the integral multiple)
N2 + 2N ⟹ 210 + 232 = 2x × k since 322 = (25)2
N2 + 2N ⟹ 210 (1 + 222) = 2x × k
This (1 + 222) is an odd number so the number is going to be 210 × multiple of the odd number
Hence the largest power of 2 can be 10 i.e. 210
Xmax = 10
The largest possible x is 10

The question is "If N and x are positive integers such that NN = 2160 and N2 + 2N is an integral multiple of 2x, then the largest possible x is [TITA]"

Hence, the answer is 10

 

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