Multiple choice

If n is a positive integer, then which of the following numbers must have a remainder of 2 when divided by any of the numbers 3, 4 and 5?

  1. 140n + 2

  2. 160n + 2

  3. 180n + 2

  4. 200n + 2

  5. 220n + 2
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C Correct answer
Explanation

The number must leave a remainder of 2 when divided by 3, 4, and 5. This means the number must be of the form LCM(3, 4, 5) * n + 2. LCM(3, 4, 5) = 60. Thus, the form is 60n + 2. Since 180n is a multiple of 60, 180n + 2 also satisfies the condition.

AI explanation

When the expression is divided by 3, 4 and 5, the constant term 2 always remains as the remainder for each case. For the entire expression to leave a remainder of 2, the variable component multiplied by n must be perfectly divisible by 3, 4 and 5. The least common multiple of 3, 4 and 5 is 60. Among the choices, the coefficient 180 is a multiple of 60, making 180n divisible by all three divisors; hence, 180n + 2 is the required expression.

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