Date | May Example questions | Marks available | 2 | Reference code | EXM.1.SL.TZ0.1 |
Level | Standard Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 0 |
Command term | Explain | Question number | 1 | Adapted from | N/A |
Question
Explain why any integer can be written in the form 4k or 4k+1 or 4k+2 or 4k+3, where k∈Z.
Hence prove that the square of any integer can be written in the form 4t or 4t+1, where t∈Z+.
Markscheme
Upon division by 4 M1
any integer leaves a remainder of 0, 1, 2 or 3. R1
Hence, any integer can be written in the form 4k or 4k+1 or 4k+2 or 4k+3, where k∈Z AG
[2 marks]
(4k)2=16k2=4t M1A1
(4k+1)2=16k2+8k+1=4t+1 M1A1
(4k+2)2=16k2+16k+4=4t A1
(4k+3)2=16k2+24k+9=4t+1 A1
Hence, the square of any integer can be written in the form 4t or 4t+1, where t∈Z+. AG
[6 marks]