If a and b are positive integers, is a^4-b^4 divisible by 4?

Question

If a and b are positive integers, is  a^4 - b^4  divisible by 4?

1) a+b is divisible by 4

2) The remainder is 2 when  a^2 + b^2  is divided by 4

vitaliyGMAT · Answer

Hi

Very good queston.

a^4 - b^4 = (a + b)(a - b)(a^2 - b^2)

(1) a+b is divisible by 4

Straightforward and sufficient.

(2) The remainder is 2 when  a^2 + b^2  is divided by 4

This is a bit tricky.

We need to be aware of the properties of remainders.

If a=r (mod c), has a remainder  r  when divided by  c , then  a^2  will have remainder  r^2  when divided by c, given c > r^2.

In our case when c=4 we can have remainders: 0, 1, 2, 3.

a = r (mod 4), b = q (mod 4)   ------>  a^2 = r^2   (mod 4) ,  b^2 = q^2   (mod 4)

We are given  r^2 + q^2 = 2 . There is only one possibility  1^2 + 1^2 = 1 + 1 = 2  (We can't get  0 + 2  or  2 + 0  because that means that  a  has a remainder  \sqrt{2}  when divided by  4  and this is impossible).

Hence a^2 - b^2 = 1 - 1 = 0 is completely divisible by  4 . Sufficient.

Answer D.

blackfish · Answer

a^4−b^4=(a+b)(a−b)(a^2+b^2)

1.  a+b  is divisible by 4

Straightforward and sufficient.

2. Let  a^2+b^2 = 4x+2  (since it gives 2 as remainder when divided by 4)

So,  a^{4}−b^{4} = (4x+2)(a+b)(a-b)  which breaks down to .. 2(2x+1)(a+b)(a-b)

Now, any number which has product of two even numbers will be divisible by 4. 
We already have 2. So we need to know if we have another even number or not.

So, 2x+1 .. always odd
  (a+b) and (a-b) both will be even if a & b both are odd or both are even .

Since  a^2+b^2  gives a remainder of 2 when divided by 4, it means the sum  sum is even .

Now, the sum will be even only if  a^2  &  b^2  both are even or both are odd.

That means, a & b both are even or both are odd (since square of odd numbers are odd and even numbers are even).

So, sufficient.

Hence, the answer is D.

hellosanthosh2k2 · Answer

(a^4 - b^4 ) % 4 = 0?

(a^2 + b ^2)(a - b)(a+b)  % 4 = 0?

Statement 1:
 (a + b) is divisible by 4 => (a + b) % 4 = 0 =>  (a^2 + b ^2)(a - b)(a+b)  is divisible by 4

Sufficient

Statement 2:
  (a^2 + b^2)  % 4 = 2, Let  (a^2 + b^2)  = 4x + 2 (where x is any integer)

To Note, since remainder is 2,  (a^2 + b^2)  must be even
 => either  a^2 hence (a) and b^2 hence(b) both are Odd 
  or 
  a^2 hence (a) and b^2 hence(b) both are Even

Substituting  (a^2 + b^2) = 4x + 2  in  (a^2 + b ^2)(a - b)(a+b) 
=>  (4x + 2)(a^2 - b^2) 
=>  4x(a^2-b^2) + 2 (a^2 - b^2)

First term, 4x(a^2-b^2) is clearly divisble by 4
Second term,  2 (a^2 - b^2)  , since we know a, b are either both odd or even, so that  (a^2 - b^2) is even =>  2 * even  => clearly divisible by 4

Sufficient

Answer (D)

itisSheldon · Answer

vitaliyGMAT 
Although you have made a good analysis, Your initial expression  a^4 - b^4 = (a + b)(a - b)(a^2 - b^2)  is not correct.
Correct form of expression:  a^4 - b^4 = (a + b)(a - b)(a^2 + b^2)

akara2500 · Answer

Can anyone please explain why the remainder = 2, a^2 + b^2 both are odd or both are even?

ganand · Answer

Hi,
Using the formula  a^{2} - b^{2} = (a+b)(a-b) , we can simplify the given expression as follows:

a^4 - b^4 = (a^{2} + b^{2})(a^{2} - b^{2}) = (a^2 + b^2)(a + b)(a - b)

(1) a+b is divisible by 4

Sufficient.

(2) The remainder is 2 when  a^2 + b^2  is divided by 4.

=>  a^{2} + b^{2} = 4k + 2 = 2(2k + 1)  (where k is an integer) --- (1)

=>  a^{2} = 4k + 2 - b^{2}

a^{2} - b^{2}  can be written as  4k + 2 - b^{2} - b^{2} = 4k + 2 - 2b^{2} = 2(2k + 1 - b^{2})  --- (2)

By using equation (1) and (2) we can write  a^4 - b^4  as follows:

a^4 - b^4 = (a^{2} + b^{2})(a^{2} - b^{2}) = 2(2k + 1) 	imes 2(2k + 1 - b^{2}) = 4 (2k + 1)(2k + 1 - b^{2})

Hence, divisible by 4. Sufficient.

Thanks.

GMATBusters · Answer

Hi  akara2500 
As per Statement 2 :
When a^2 + b^2 is divisible by 4, remainder =2.
Hence a^2+ b^2 = 4k+2, thats is even.
Hence either both a,b are even or both a,b are odd.
If both are even, a =2p, b =2q. Hence a^4- b^4 = 16(p^4-q^4). Hence divisible by 4.
If both are odd, a=(2m+1), b=(2n+1)
Now a^2-b^2=4(m^2-n^2+m-n). Hence divisible by 4. Also a^4-b^4= (a^2-b^2)(a^2+b^2). Hence divisible by 4.
Statement 2 is also sufficient.

Answer D

Can anyone please explain why the remainder = 2, a^2 + b^2 both are odd or both are even?

SSM700plus · Answer

a^4−b^4=(a+b)(a−b)(a^2+b^2)  ---------1

1. 1st statement is very simple and straight forward. And it is sufficient.

2.  Let see how we will do for 2nd Statement.

Let  a^2+b^2=4x+2  -----------2

where x is the quotient and 2 is the remainder.

Using 2 in 1 we get  a^4−b^4=(4x+2)(a+b)(a−b)

Which breaks down to  a^4−b^4=2(2x+1)(a+b)(a−b)

We have to find out if eq 1 is divisible by 4. 
Product of any two even numbers will be divisible by 4.

We have one even number- 2. So we need to find out if any of the expression (2x+1), (a+b) & (a-b) will give us even number?

We know that 2x+1 is always odd. NOTE - it is general representation of ODD numbers.

In what scenario (a+b) & (a-b) will give us odd/even.
1) If a and b is odd. Then both (a+b) and (a-b) will be even.
2) If a and b is even. Then both (a+b) and (a-b) will be even.

But how can we decide if a & b is even or odd?

Since  a^2+b^2  gives a remainder of 2 when divided by 4, it means the sum is even.

From here we have to backtrack with logic to determine if a & b is odd or even.

Again, sum will be even if both the numbers are odd or both the numbers are even.

There are two scenario both leading to common results:-
1) If the square of a number is odd. then the number itself will be odd.
2) Similarly, if square of the numbers are even then, the numbers itself will be even.

From scenario 1 :- Since sum of expression  a^2+b^2  is even, let's assume both  a^2  and  b^2  are odd. Then it implies both a & b are odd. Also the expression (a+b) & (a-b) will be even.

From scenario 1 :- Since sum of expression  a^2+b^2  is even, let's assume both  a^2  and  b^2  are even. Then it implies both a & b are even. Also the expression (a+b) & (a-b) will be even.

Hence the expression  a^4−b^4  is divisible by 4. So, sufficient.

Hence, the answer is D.

Hope this helps!!
Thanks

GMATBusters · Answer

Official Explanation: 
 a^4-b^4=(a-b)(a+b)(a^2+b^2) 
 
Statement 1 :  
a+b is divisible by 4, a+b = 4k
Since,  a^4-b^4=(a-b)(a+b)(a^2+b^2) , 
we get,  a^4-b^4=4k(a-b)(a^2+b^2) 
Hence it is divisible by 4, 
 SUFFICIENT

Statement 2 :  
 a^2+b^2  , when divided by 4 gives 2 as remainder, Hence  a^2+b^2  is Even.
It means either both a and b are EVEN or ODD.
When both are Even, a = 2m, b = 2n:  a^4-b^4 = 4(m^4-n^4) , divisible by 4
When both are oddm a = 2m +1, b = 2n+1: 
  a^4-b^4=(a-b)(a+b)(a^2+b^2) 
= 4(m-n)(m+n+1)(a^2+b^2) , hence divisible by 4

SUFFICIENT

Answer D

pk123 · Answer

a4−b4 = (a+b)*(a-b) (a2+b2)

Option A: a+b is divisible by 4
since a and b are integers, it definitely answers that a4-b4 will be divisible by 4

2) The remainder is 2 when a2+b2 is divisible by 4
which means a2+b2=. 4k +2

so it becomes (a+b)(a-b) (4k+2)
can't say that it is divisible by 4

A is the answer

Manvi_nagpal · Answer

A

hargun3045 · Answer

a^4 - b^4 = (a-b)(a+b)(a^2-b^2)

St 1 says a+4 div by 4 - sufficient

St 2 says a^2+b^2mod 4 = 2 (remainder 2)
which means a and b are either both odd, or both even
Both cases imply a+b is even Hence product of two even numbers is div by 4

D

Gladiator59 · Answer

a^4-b^4 can be factorized as (a^2-b^2)*(a^2+b^2)... (a-b)*(a+b)*(a^2+b^2)

St.1 says a+b is divisible by 4. Hence sufficient.

St.2 says remainder is 2 when a^2+b^2 is divided by 4.. hence it is divisible by 2. It could have both even or both odd parts adding up to an even total... Hence either ways a+b and a-b will be even. Therefore the whole will be divisible by 4.

St2. Is also sufficient.

Hence option (d) is correct.

Best,
G

Posted from my mobile device

rever08 · Answer

Answer D

u1983 · Answer

a^4-b^4=(a^2+b^2)(a+b)(a-b)

1) suff
2)Suff
(a^2+b^2) div 4..... rem 2 .....===> (a^2+b^2) is divisible by 2 not by 4
case 1 : so a, b both even .... then (a+b) & (a-b) both div by 2
case 2 : so a, b both odd .... then (a+b) & (a-b) both div by 2

Hence ans D

OJA · Answer

This is a nice question.

anirchat · Answer

Agreed ! The intuition is simply great.

If a and b are positive integers, is a^4-b^4 divisible by 4?

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