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Which of the following is a complete list of the positive factors of 60?Read the answer and worked solution →Which statement about even and odd integers is **false**?Read the answer and worked solution →When the integer $-32$ is divided by $3$, the quotient $q$ and remainder $r$ (with $0 \le r < 3$) satisfy:Read the answer and worked solution →The prime factorization of $1{,}155$ is:Read the answer and worked solution →

Worked practice questions

Which of the following is a complete list of the positive factors of 60?

  1. 1, 2, 3, 4, 5, 6, 10, 15, 30, 60
  2. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  3. 2, 3, 4, 5, 6, 10, 12, 15, 20, 30
  4. 1, 2, 3, 5, 6, 10, 12, 15, 20, 60
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Correct answer: B. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Pair the divisors: $1 \times 60$, $2 \times 30$, $3 \times 20$, $4 \times 15$, $5 \times 12$, $6 \times 10$. That gives 12 positive factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Option A misses 4, 12 and 20. Option C omits 1 and 60 (every positive integer has itself and 1 as factors). Option D drops 4.

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Which statement about even and odd integers is **false**?

  1. The sum of two odd integers is even.
  2. The product of an even integer and an odd integer is even.
  3. The product of two odd integers is odd.
  4. The sum of an even integer and an odd integer is even.
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Correct answer: D. The sum of an even integer and an odd integer is even.

Even + odd is always odd, not even. Example: $4 + 3 = 7$. The other three are standard parity facts: - odd + odd = even (e.g. $3 + 5 = 8$). - even $\times$ odd = even (any multiple of an even number is even). - odd $\times$ odd = odd (no factor of 2 is introduced).

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When the integer $-32$ is divided by $3$, the quotient $q$ and remainder $r$ (with $0 \le r < 3$) satisfy:

  1. $q = -10$, $r = -2$
  2. $q = -10$, $r = 2$
  3. $q = -11$, $r = 1$
  4. $q = -11$, $r = -1$
Show answer and solution

Correct answer: C. $q = -11$, $r = 1$

By definition the remainder must satisfy $0 \le r < d$ where $d$ is the (positive) divisor. So $r$ cannot be negative — that rules out A and D. We need the **greatest** multiple of $3$ that is $\le -32$. Try $q = -10$: $-10 \times 3 = -30$, but $-30 > -32$ so $-30$ is **not** $\le -32$. Try $q = -11$: $-11 \times 3 = -33 \le -32$ ✓. Remainder $= -32 - (-33) = 1$. Check: $-32 = (-11)(3) + 1$ ✓.

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