swerve wrote:If m and n are positive integers such that m>n, what is the remainder when m^2-n^2 is divided by 21?
1) The remainder when (m+n) is divided by 21 is 1.
2) The remainder when (m-n) is divided by 21 is 1.
.
Given: m and n are positive integers such that m > n
Target question: What is the remainder when m² - n² is divided by 21?
Statement 1: The remainder when (m + n) is divided by 21 is 1
There are several values of m and n that satisfy statement 1. Here are two:
Case a: m = 12 and n = 10. This means m + n = 12 + 10 = 22, and 22 divided by 21 leaves remainder 1. In this case, m² - n² = 12² - 10² = 144 - 100 = 44.
When we divide 44 by 21, we get 2 with remainder 2. So, the answer to the target question is
when m² - n² is divided by 21, the remainder is 2
Case b: m = 13 and n = 9. This means m + n = 13 + 9 = 22, and 22 divided by 21 leaves remainder 1. In this case, m² - n² = 13² - 9² = 169 - 81 = 88.
When we divide 88 by 21, we get 4 with remainder 4. So, the answer to the target question is
when m² - n² is divided by 21, the remainder is 4
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: The remainder when (m - n) is divided by 21 is 1
There are several values of m and n that satisfy statement 2. Here are two:
Case a: m = 5 and n = 4. This means m - n = 5 - 4 = 1, and 1 divided by 21 leaves remainder 1. In this case, m² - n² = 5² - 4² = 25 - 16 = 9.
When we divide 9 by 21, we get 0 with remainder 9. So, the answer to the target question is
when m² - n² is divided by 21, the remainder is 9
Case b: m = 4 and n = 3. This means m - n = 4 - 3 = 1, and 1 divided by 21 leaves remainder 1. In this case, m² - n² = 4² - 3² = 16 - 9 = 7.
When we divide 7 by 21, we get 0 with remainder 7. So, the answer to the target question is
when m² - n² is divided by 21, the remainder is 7
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1 tells us that the remainder is 1 when (m + n) is divided by 21
In other words, m+n is
1 greater than some multiple of 21.
So, we can write:
m+n = 21k + 1 (for some integer k)
Statement 2 tells us that the remainder is 1 when (m - n) is divided by 21
In other words, m-n is
1 greater than some multiple of 21.
So, we can write:
m-n = 21j + 1 (for some integer j)
Now recognize that we can factor m² - n²
We get: m² - n² = (
m + n)(
m - n)
= (
21k + 1)(
21j + 1)
= 21²mn + 21k + 21j + 1
= 21(21mn + k + j) + 1
Since 21(21mn + k + j) is definitely a multiple of 21, we can conclude that 21(21mn + k + j) + 1 is
1 greater than some multiple of 21.
In other words, m² - n² is
1 greater than some multiple of 21.
So,
when m² - n² is divided by 21, the remainder is 1
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
