The difference between the squares of two consecutive even integers will always be divisible by which of the following?
(A) 2
(B) 3
(C) 4
(D) 5
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (C) only
Write the two consecutive even integers as \(2n\) and \(2n+2\). Then the difference of their squares is 4 times an odd number:
\((2n+2)^2=4n^2+8n+4\) and \((2n)^2=4n^2\)
The difference \(=8n+4=4(2n+1)\)
\(2n+1\) is odd, so the difference is always a multiple of 4, but never of 8
(A) 2 Correct. Every multiple of 4 is also a multiple of 2.
(B) 3 Incorrect. For 4 and 6, \(36-16=20\), which is not divisible by 3.
(C) 4 Correct. The difference is \(4(2n+1)\), always a multiple of 4.
(D) 5 Incorrect. For 2 and 4, \(16-4=12\), which is not divisible by 5.