Factors and Multiples - SSAT Middle Level Quantitative

1. Identify whether $7$ is a factor of $42$.

   Yes, because $42=7\times 6$. Dividing 42 by 7 gives an integer, confirming 7 as a factor.

2. Identify whether $8$ is a factor of $54$.

   No, because $54$ is not divisible by $8$. 54 divided by 8 yields a non-integer, so 8 does not divide 54 evenly.

3. What is the divisibility rule for $6$?

   Divisible by both $2$ and $3$. Since 6 = 2 × 3, a number must satisfy both divisibility rules to be a multiple of 6.

4. Identify the greatest factor of $45$ that is less than $10$.

   $9$. Among factors of 45 (1, 3, 5, 9, 15, 45), 9 is the largest below 10.

5. Identify all positive factors of $12$.

   $1,2,3,4,6,12$. The positive factors are all whole numbers that divide 12 evenly, listed in ascending order.

6. What is the next multiple of $8$ after $56$?

   $64$. After 56 = 8 × 7, the next is 8 × 8, following the sequence of multiples.

7. Which numbers have exactly two positive factors?

   Prime numbers. Primes are greater than 1 and divisible only by 1 and themselves, yielding exactly two factors.

8. What is the divisibility rule for $9$?

   The sum of digits is divisible by $9$. Divisibility by 9 requires the sum of digits to be a multiple of 9, similar to the rule for 3.

9. Identify whether $1$ is prime, composite, or neither.

   Neither prime nor composite. The number 1 has only one positive divisor, failing the criteria for both prime and composite.

10. What is the definition of a factor of a whole number $n$?

    A whole number that divides $n$ with remainder $0$. A factor evenly divides n, resulting in an integer quotient and zero remainder.

11. What is the divisibility rule for $10$?

    The last digit is $0$. Ending in 0 means the number is divisible by both 2 and 5, hence by 10.

12. Identify whether $36$ is a multiple of $9$.

    Yes, because $36=9\times 4$. 36 is obtained by multiplying 9 by 4, establishing it as a multiple.

13. What does it mean if $a$ is a factor of $b$ in divisibility notation?

    $a \mid b$. The notation signifies that a divides b exactly, with b being a multiple of a.

14. What does it mean if $a$ is a multiple of $b$ in divisibility notation?

    $b \mid a$. The notation indicates that b divides a exactly, meaning a is an integer multiple of b.

15. What is the divisibility rule for $5$?

    The last digit is $0$ or $5$. A number ends in 0 or 5 when it is a multiple of 5, per the base-10 system.

16. What is the divisibility rule for $3$?

    The sum of digits is divisible by $3$. Divisibility by 3 occurs when the digital root or sum of digits is a multiple of 3.

17. What is the least positive multiple of $7$?

    $7$. The least positive multiple is 7 × 1, as positive multiples start from the number itself.

18. Which numbers have more than two positive factors?

    Composite numbers. Composites are greater than 1 and not prime, having at least one divisor other than 1 and themselves.

19. What is the divisibility rule for $2$?

    The last digit is even: $0,2,4,6,8$. A number is divisible by 2 if its units digit is even, indicating it is a multiple of 2.

20. What is the definition of a multiple of a whole number $n$?

    Any number of the form $n \times k$ for whole $k$. Multiples are obtained by multiplying n by any whole number k, forming the sequence 0, n, 2n, etc.

21. Which option is a common factor of $24$ and $36$: $5$, $6$, or $7$?

    $6$. 6 divides both 24 and 36 evenly, unlike 5 and 7, making it a common factor.

22. What are the two guaranteed positive factors of every whole number $n>0$?

    $1$ and $n$. Every positive integer n is divisible by 1 and by n itself, ensuring these as factors.

23. Identify all positive factors of $18$.

    $1,2,3,6,9,18$. The positive factors are all divisors of 18 that yield integer quotients with no remainder.

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