If c and d are the squares of odd integers, which of the following is

Question

If c and d are the squares of odd integers, which of the following is a possible value of c – d?

A) 1008
B) 1018
C) 1022
D) 1030
E) 1038

GMATinsight · Accepted Answer

c and d are the squares of odd integers

Let,  c = (2p+1)^2  and  d = (2q+1)^2

i.e.  c-d = (2p+1)^2 - (2q+1)^2 = 4(p^2-q^2) - 4(p-q)

i.e. Result Must be divisible by 4

Divisibility test of 4: Last two digits of the number must be divisible by 4

A) 10 08   i.e. 08 is Divisible by 4 
B) 10 18  18 is NOT divisible by 4
C) 10 22  22 is NOT divisible by 4
D) 10 30  30 is NOT divisible by 4
E) 10 38  38 is NOT divisible by 4

ANswer: Option A

BrentGMATPrepNow · Answer

SOLUTION #1 : 
All odd integers can be rewritten as 2k + 1, where k is an integer. 
So, if c is the square of an odd integer, we can write: c = (2k + 1)2 for some integer k. 
Likewise, we can write: d = (2j + 1)2 for some integer j.

So, c - d = (2k + 1)² - (2j + 1)²
Expand: c – d = (4k² – 4k + 1) - (4j² – 4j + 1)
Simplify: c – d = 4k² – 4k - 4j² – 4j
Factor to get: c – d = 4(k² – k - j² – j)
This tells us that c – d is a multiple of 4
Since answer choice A is the only multiple of 4, it must be the correct answer.

SOLUTION #2 :
If c and d are the squares of odd integers, we can let c = x² and let d =y²(where x and y are odd integers)

Now take: c - d
And rewrite as: x² – y²
Factor: (x + y) (x - y)

Since x and y are both odd, we can rewrite the above expression as follows: (odd + odd) (odd - odd)
Simplify to get: (some EVEN integer)(some other EVEN integer)

If an integer is EVEN, we can rewrite it as 2k, where k is some integer. 
So, we can take (some EVEN integer)(some other EVEN integer) and rewrite it as: (2j)(2k), where j and k are integers. 
Finally we can rewrite (2j)(2k) as 4jk

We’ve now taken c - d and rewritten it as 4jk, where j and k are integers. 
Since  4 jk is a multiple of 4, it must be the case that c - d is a multiple of 4.

When we check the answer choices, we see that only answer choice A is a multiple of 4. 
Answer: A

SOLUTION #3 : 
Even if you didn't identify an algebraic solution, you can still test some possible values and  look for a pattern . 
Since c and d are the squares of odd integers, it could be the case that c = 9² = 81, and d =7² = 49, in which case c - d = 81 - 49 =  32 
Or it could be the case that c = 9² = 81, and d =5² = 25, in which case c - d = 81 - 25 =  56 
Or it could be the case that c = 9² = 81, and d =3² = 9, in which case c - d = 81 - 9 =  72 
Or it could be the case that c = 9² = 81, and d =1² = 1, in which case c - d = 81 - 1 =  80 
Or it could be the case that c = 11² = 121, and d =9² = 81, in which case c - d = 121 - 81 =  40 
At this point, we might recognize that  32, 56, 72, 80  and  40  are all divisible by 4, in which case it could be true that the correct answer is also divisible by 4. 
Answer: A

Aside: As you might guess, SOLUTION #3 isn't the most precise way to approach this question, but it still gives you a reasonable chance of identifying the correct answer

Archit3110 · Answer

we know that difference of any two odd integers would give us an even integer value
and possible unit digits of the square of odd integers are ; 1,9,5 and its difference would give us units digits as 0,4,8
we can eliminate option c
also point to notice is that for remainder 0,4,8 the LCM is 4 ; therefore the answer option should be divisible by 4
which in this case is only option A ; 1008

AkshdeepS · Answer

I did this question by hit and trial method.

As all the numbers are above 1000, I tried to find out the first largest square of odd number above 1000, and that is  33 * 33 = 1089

Now, our task is to the check smaller squares of odd numbers .

3*3 = 9  ;  1089 - 9 = 1080  (No match)

Choices are much smaller than 1080

So I tried  9*9 = 81

1089 - 81 = 1008  (Matched with A)

As there can be only one right answer, tick choice A and move on.

Hence A.

iammohakjain · Answer

Posted from my mobile device

rafdezg · Answer

I think that when you are expanding (2k + 1)^2 , there must be an error, all terms should be positive when you expand that quadratic property. I see you have -4k inside the parantheses of the expansion. Let me know if I'm not seeing something.

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If c and d are the squares of odd integers, which of the following is

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