We define “A mod n” as the remainder when a positive integer A is divi

Target question: What is the value of A² mod 4?
In other words, what is the remainder when A² is divided by 4?

Statement 1: A mod 2=1
In other words, When A is divided by 2, the remainder is 1
ASIDE: There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

So, in this case, we can write: A = 2k + 1 (where k is some integer)
If A = 2k + 1, then A² = (2k + 1)² = 4k² + 4k + 1 = 4(k² + k) + 1
In other words, A² = 4(some integer) + 1
Since A² is 1 GREATER THAN some multiple of 4, we can conclude that the remainder is 1 when A² is divided by 4
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: A mod 8=1
In other words, When A is divided by 8, the remainder is 1
So, in this case, we can write: A = 8k + 1 (where k is some integer)
If A = 8k + 1, then A² = (8k + 1)² = 64k² + 16k + 1 = 4(16k² + 4k) + 1
In other words, A² = 4(some integer) + 1
Since A² is 1 GREATER THAN some multiple of 4, we can conclude that the remainder is 1 when A² is divided by 4
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer:
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