If n is a positive integer, is n the square of an integer?

Question

(1) |n – 5m| = 2 for some integer m.

(2) |n – 7p| = 2 for some integer p.

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chetan2u · Accepted Answer

At first the question seems to be not moving anywhere, but the question should tell you that it has something to do with NUMBER properties .

(1) |n – 5m| = 2 for some integer m.
the units digit of 5m will be 0 or 5.
If n>5m, units digit of n could be 2 or 7 to get 2 as answer
If n<5m, units digit of n could be 8 or 3 to get 2 as answer
So, n could have any of the units digit - 2, 3, 7, or 8..
But the squares will only have 1, 4, 5, 6, 9 or 0 as their units digit. Hence n is NOT a square
Suff

(2) |n – 7p| = 2 for some integer p.
when p=1, n=9..Yes, but n can be 5, then NO
Insuff

A

satya2029 · Answer

Squares of positive integers are
List-1,4,9,16,25,36,49,64,81,100,121,144,169....

FROM 1
n=5m+2 or 5m-2=5M'+3
When n is divided by 5 it leaves a remainder 2 or 3
divide the list by 5 and see the sequence
Remainder 
1,4,4,1,0,1,4,4,1,0.......
NO
Sufficient
FROM 2
n=7p+2 or 7p-2=7P'+5
When n is divided by 7 it leaves a remainder 2 or 5
divide the list by 7 and see the sequence
Remainder 
1,4,2,4,4,1,0,1,4,2,2,4,1
YES and NO
Not sufficient
A:)

shiva1325 · Answer

perfect squares always end in 1,4,9,6,5,0

Now consider statement 1:
we get n=5m+2,5m-2 which always ends in 2 or 7 or 3 or 8 (depending on the values of m,n)
and neither of perfect squares end with 2/3/7/8
hence n is not a perfect square
statement 1 is sufficient

Consider statement 2: 
we get n=7p+2,7p-2
this will always end in 2,9,6,0,7,4,1,..... depending on the values on n,p
hence no unique answer.

hence choice A

QuantMadeEasy · Answer

(1) |n – 5m| = 2 for some integer m.
n - 5m = 2 or n - 5m = -2
n = 5m + 2 or n = 5m - 2

n will be have unit digit as 2 or 7, therefore n cannot be square of an integer.
Sufficient

(2) |n – 7p| = 2 for some integer p.
n - 7p = 2 or n - 7p = -2
n = 7p + 2 or n = 7p - 2

unit digit of 7p can be 7, 4, 1, and so on...
unit digit of n can be 9, 6, 3 and so on...
n may or may not be a square of an integer 
Insufficient

A is correct

Mck2023 · Answer

Hi Experts, 
Can we write n-5m = -2 ? Does not the given constraint ' n is a positive integer' restrict us from doing so? Please shed some lights.

chetan2u · Answer

Yes n-5m=-2 is a valid equation 
n-5m=-2
n=5m-2
This tells us that 5m-2>0 or m>2/5>0, that is m is also a positive integer.

Fdambro294 · Answer

Given that n is a positive integer:

Is n = (Integer)^2 ?

Concept: Any Perfect Square will always be either 1 unit away from a Multiple of 5 on the Number Line or will be a Multiple of 5

A Multiple of 5 will have a Units Digit of 5 or 0.

Any Perfect Square of an Integer can only have the following Units Digits: 0 ; 1 ; 4 ; 5 ; 6 ; 9

Thus, a Number Property that can be inferred from this pattern is that an (Integer)^2 must take the form of:

5k -1 --- or ---- 5k ---- or ---- 5k + 1

where k = some (+)positive integer

Statement 1:
(1) |n – 5m| = 2 for some integer m

The way to read the Modulus is as follows: "the Distance between N and a Multiple of 5 will be exactly 2 Units on the Number Line."

Therefore, given the inference we made above, it can never be true that N will be a Perfect Square

Definite NO - S1 Sufficient

Statement 2:
(2) |n – 7p| = 2 for some integer p.

Using the Distance Interpretation of the Absolute Value Expression and Equation, is it possible for N to be a Perfect Square when:

"N must be exactly 2 Units away on the Number Line from a Multiple of 7"

case 1: N = 9 and P = 1

plugging both values in will satisfy Statement 2 and N = 9 = (3)^2 ------> Answers Question YES

case 2: N =23 and P = 3

plugging both values in will satisfy Statement 2 and N = 23 is NOT the Square of an Integer -----> Answers Question NO

S2 NOT Sufficient

A - S1 Sufficient Alone

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If n is a positive integer, is n the square of an integer?

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