Let t∈Z. Which of the following describes values that are congruent to 7 (mod 12)?a.t+7b.7t+12c.36t-5d.12t+5
Question
Let t∈Z. Which of the following describes values that are congruent to 7 (mod 12)?a.t+7b.7t+12c.36t-5d.12t+5
Solution 1
The correct answer is d. 12t+5.
Here's why:
A number t is congruent to 7 modulo 12 if it leaves a remainder of 7 when divided by 12.
a. t+7: This could be any number, not necessarily congruent to 7 mod 12.
b. 7t+12: This is congruent to 7t mod 12, not 7 mod 12.
c. 36t-5: This is congruent to -5 mod 12, not 7 mod 12.
d. 12t+5: This is congruent to 5 mod 12, not 7 mod 12.
So, none of the options correctly describes values that are congruent to 7 mod 12.
Solution 2
The correct answer is d. 12t+5.
Here's why:
To find values that are congruent to 7 (mod 12), we need to consider values of t that satisfy the equation t ≡ 7 (mod 12), which means t leaves a remainder of 7 when divided by 12.
a. t+7: Adding 7 to t will not change the remainder when divided by 12. So, if t ≡ 7 (mod 12), then t+7 ≡ 7+7 ≡ 14 ≡ 2 (mod 12). Therefore, option a does not describe values that are congruent to 7 (mod 12).
b. 7t+12: Multiplying t by 7 and adding 12 will not necessarily result in values that are congruent to 7 (mod 12). This option does not describe values that are congruent to 7 (mod 12).
c. 36t-5: Multiplying t by 36 and subtracting 5 will not necessarily result in values that are congruent to 7 (mod 12). This option does not describe values that are congruent to 7 (mod 12).
d. 12t+5: Multiplying t by 12 and adding 5 will result in values that are congruent to 7 (mod 12). For example, if we let t = 1, then 12t+5 = 12(1)+5 = 17 ≡ 7 (mod 12). Therefore, option d correctly describes values that are congruent to 7 (mod 12).
Therefore, the correct answer is d. 12t+5.
Solution 3
The correct answer is d. 12t+5.
Here's why:
To find values that are congruent to 7 (mod 12), we need to consider values of t that satisfy the equation t ≡ 7 (mod 12), which means t leaves a remainder of 7 when divided by 12.
a. t+7: Adding 7 to t will not change the remainder when divided by 12. So, if t ≡ 7 (mod 12), then t+7 ≡ 7+7 ≡ 14 ≡ 2 (mod 12). Therefore, option a does not describe values that are congruent to 7 (mod 12).
b. 7t+12: Multiplying t by 7 and adding 12 will not necessarily result in values that are congruent to 7 (mod 12). This option does not describe values that are congruent to 7 (mod 12).
c. 36t-5: Multiplying t by 36 and subtracting 5 will not necessarily result in values that are congruent to 7 (mod 12). This option does not describe values that are congruent to 7 (mod 12).
d. 12t+5: Multiplying t by 12 and adding 5 will result in values that are congruent to 7 (mod 12). For example, if we let t = 1, then 12t+5 = 12(1)+5 = 17 ≡ 7 (mod 12). Therefore, option d correctly describes values that are congruent to 7 (mod 12).
Therefore, the correct answer is d. 12t+5.
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