6. Factors and Multiples

Practice Questions – 

1. Check if $3245$ is divisible by $5$ and $9$.

Step 1: Divisibility by $5$:

• Rule: A number is divisible by $5$ if its last digit is $0$0 or $5$.
• Last digit of $3245$ is $5$.
  Hence, $3245$is divisible by $5$.

Step 2: Divisibility by $9$:

• Rule: A number is divisible by $9$ if the sum of its digits is divisible by $9$.
• Sum of digits: $3+2+4+5 = 14$.
  $14 div 9$ leaves a remainder of $5$.
  Hence, $3245$ is not divisible by $9$.

Final Answer: $3245$ is divisible by $5$ but not by $9$.

2. A number $n = 3^3 times 2^2 times 7$. Is $n$ divisible by $6$ and $9$?

Step 1: Divisibility by $6$:

• Rule: $6 = 2 times 3$. $n$ must contain at least one factor of $2$ and $3$.
  $n = 3^3 times 2^2 times 7$
  $n$ has $2$ and $3$ as factors.
  Hence, $n$ is divisible by $6$.

Step 2: Divisibility by $9$:

• Rule: $9 = 3^2$. $n$ must contain at least $3^2$ as a factor.
  $n = 3^3 times 2^2 times 7$.
  $n$ contains $3^3$, so it is divisible by $9$.

Final Answer: $n$n is divisible by both $6$6 and $9$9.

3. Test divisibility of $246810$ by $2, 5, text{and } 11$.

Step 1: Divisibility by $2$:

• Rule: A number is divisible by $2$ if its last digit is even.
  Last digit of $246810$ is $0$ (even).
  Hence, $246810$ is divisible by $2$.

Step 2: Divisibility by $5$:

• Rule: A number is divisible by $5$ if its last digit is $0$ or $5$.
  Last digit of $246810$ is $0$.
  Hence, $246810$ is divisible by $5$.

Step 3: Divisibility by $11$:

• Rule: A number is divisible by $11$ if the difference between the sum of digits in odd positions and even positions is divisible by $11$.
  Digits of $246810$: $2, 4, 6, 8, 1, 0$2,4,6,8,1,0.
  Odd positions: $2 + 6 + 1 = 9$.
  Even positions: $4 + 8 + 0 = 12$.
  Difference: $|9 – 12| = 3$.
  $3 div 11$ leaves a remainder.
  Hence, $246810$246810 is not divisible by $11$.

Final Answer: $246810$ is divisible by $2$ and $5$, but not by $11$.

4. Find all the factors of $48$.

Step 1: Definition of factors:
Factors of a number are integers that divide the number exactly without leaving a remainder.

Step 2: Test for divisibility from $1$ to $sqrt{48}$:

$$sqrt{48} approx 6.93$$

We test divisibility for numbers from $1$ to $6$:

• $48 div 1 = 48$ ($1$ and $48$ are factors)
• $48 div 2 = 24$ ($2$ and $24$ are factors)
• $48 div 3 = 16$ ($3$ and $16$ are factors)
• $48 div 4 = 12$($4$ and $12$ are factors)
• $48 div 6 = 8$ ($6$ and $8$ are factors)

Step 3: List all factors:

$$text{Factors of } 48 = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.$$

5. List the factors of $72$ between $1$ and $36$.

Step 1: Find all factors of $72$:

• Prime factorization: $72 = 2^3 times 3^2$.
• Using the factors method:

$$text{Factors of } 72 = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}.$$

Step 2: Select factors $leq 36$:

$${1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36}.$$

6. Which of the following numbers are factors of $90$: $2, 5, 11, 15$?

Step 1: Divide $90$ by each number:

• $90 div 2 = 45$ ($2$ is a factor).
• $90 div 5 = 18$ ($5$ is a factor).
• $90 div 11 neq text{an integer}$ ($11$ is not a factor).
• $90 div 15 = 6$ ($15$ is a factor).

Answer: $2, 5, 15$ are factors of $90$.

7. A number $n$ is divisible by $6$ and $9$. List any three factors of $n$.

Step 1: Definition of divisibility:
If $n$ is divisible by $6$ and $9$, then $n$ is divisible by their LCM.

$$text{LCM of } 6 text{ and } 9 = 18.$$

Step 2: List factors of $18$:

$${1, 2, 3, 6, 9, 18}.$$

Step 3: Any three factors of $n$:

$${2, 3, 6}.$$

8. Find the smallest factor of $81$ greater than $1$.

Step 1: Prime factorization of $81$:

$$81 = 3^4.$$

Step 2: Smallest factor $> 1$: The smallest prime factor of $81$ is $3$.

Answer: $3$.

9. Determine if $63$63 is a factor of $189$189.

Step 1: Check divisibility:
Divide $189$189 by $63$63:

$$189 div 63 = 3.$$189÷63=3.

Since the result is an integer, $63$63 is a factor of $189$189.

7. Find the largest factor of $56$56 that is less than $30$30.

Step 1: List all factors of $56$56:

$${1, 2, 4, 7, 8, 14, 28, 56}.$${1,2,4,7,8,14,28,56}.

Step 2: Identify factors $< 30$<30:

$${1, 2, 4, 7, 8, 14, 28}.$${1,2,4,7,8,14,28}.

Answer: The largest factor is $28$28.

8. If $k$k is a factor of $120$120 and $k < 10$k<10, list all possible values of $k$k.

Step 1: List all factors of $120$120:

$${1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120}.$${1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}.

Step 2: Select factors $< 10$<10:

$${1, 2, 3, 4, 5, 6, 8}.$${1,2,3,4,5,6,8}.

Answer: $k = {1, 2, 3, 4, 5, 6, 8}$k={1,2,3,4,5,6,8}.

Multiples

9. List the first five multiples of $8$8.

Answer:

$$8 times 1 = 8, , 8 times 2 = 16, , 8 times 3 = 24, , 8 times 4 = 32, , 8 times 5 = 40.$$8×1=8,8×2=16,8×3=24,8×4=32,8×5=40. $$text{First five multiples of } 8: {8, 16, 24, 32, 40}.$$First five multiples of 8:{8,16,24,32,40}.

10. What is the smallest 2-digit multiple of $7$7?

Step 1: Smallest 2-digit number is $10$10.

Step 2: Divide $10$10 by $7$7:

$$10 div 7 = 1.43 , text{(round up to (2))}.$$10÷7=1.43(round up to 2).

Step 3: Multiply $7 times 2 = 14$7×2=14.

Answer: $14$14.

11. Determine whether $125$125 is a multiple of $5$5.

Step 1: Check divisibility by $5$5:

$$125 div 5 = 25.$$125÷5=25.

Since the result is an integer, $125$125 is a multiple of $5$5.

12. Find the smallest multiple of $12$12 greater than $50$50.

Step 1: Divide $50$50 by $12$12:

$$50 div 12 = 4.167.$$50÷12=4.167.

Step 2: Round up to the next integer ($5$5):

$$12 times 5 = 60.$$12×5=60.

Answer: $60$60.

13. If $x$x is a multiple of $6$6, find the smallest value of $x$x greater than $100$100.

Step 1: Divide $100$100 by $6$6:

$$100 div 6 = 16.67.$$100÷6=16.67.

Step 2: Round up to the next integer ($17$17):

$$6 times 17 = 102.$$6×17=102.

Answer: $102$102.

14. List all multiples of $4$4 between $30$30 and $60$60.

Step 1: Divide $30$30 by $4$4:

$$30 div 4 = 7.5 , text{(round up to (8))}.$$30÷4=7.5(round up to 8).

Step 2: Divide $60$60 by $4$4:

$$60 div 4 = 15.$$60÷4=15.

Step 3: List multiples:

$$4 times 8 = 32, , 4 times 9 = 36, , 4 times 10 = 40, , 4 times 11 = 44, , 4 times 12 = 48, , 4 times 13 = 52, , 4 times 14 = 56.$$4×8=32,4×9=36,4×10=40,4×11=44,4×12=48,4×13=52,4×14=56.

Answer: ${32, 36, 40, 44, 48, 52, 56}${32,36,40,44,48,52,56}.

15. How many multiples of $9$9 are there between $50$50 and $150$150?

Step 1: Find the smallest multiple of $9$9 greater than $50$50:

$$50 div 9 = 5.56 , text{(round up to (6))}.$$50÷9=5.56(round up to 6). $$9 times 6 = 54.$$9×6=54.

Step 2: Find the largest multiple of $9$9 less than $150$150:

$$150 div 9 = 16.67 , text{(round down to (16))}.$$150÷9=16.67(round down to 16). $$9 times 16 = 144.$$9×16=144.

Step 3: Count multiples from $6$6 to $16$16:

$$16 – 6 + 1 = 11.$$16−6+1=11.

Answer: $11$11 multiples.

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Prime Factors – Detailed Solutions

16. Find the prime factorization of $84$84.

Step 1: Divide $84$84 by the smallest prime number ($2$2):

$$84 div 2 = 42.$$84÷2=42.

Step 2: Continue dividing by $2$2 until it no longer divides evenly:

$$42 div 2 = 21 quad (text{Stop, as (21) is not divisible by (2)}).$$42÷2=21(Stop, as 21 is not divisible by 2).

Step 3: Move to the next smallest prime ($3$3):

$$21 div 3 = 7.$$21÷3=7.

Step 4: Since $7$7 is prime, stop.

Final Prime Factorization:

$$84 = 2^2 times 3 times 7.$$84=22×3×7.

17. Find the prime factors of $150$150.

Step 1: Divide $150$150 by the smallest prime number ($2$2):

$$150 div 2 = 75.$$150÷2=75.

Step 2: $75$75 is odd, so divide by the next smallest prime ($3$3):

$$75 div 3 = 25.$$75÷3=25.

Step 3: $25$25 is not divisible by $3$3, so divide by $5$5:

$$25 div 5 = 5 quad (text{Keep dividing by (5)}).$$25÷5=5(Keep dividing by 5).

Step 4: Divide $5$5 by $5$5:

$$5 div 5 = 1.$$5÷5=1.

Final Prime Factorization:

$$150 = 2 times 3 times 5^2.$$150=2×3×52.

18. Determine if $81$81 is a prime number or composite. If composite, find its prime factors.

Step 1: Definition of prime number:
A prime number has exactly two factors ($1$1 and itself).

Step 2: Check divisibility:
$81$81 is divisible by $3$3:

$$81 div 3 = 27.$$81÷3=27.

Since $81$81 has factors other than $1$1 and itself, it is composite.

Step 3: Prime factorization of $81$81:

$$81 = 3^4.$$81=34.

19. List all prime factors of $210$210.

Step 1: Divide $210$210 by $2$2:

$$210 div 2 = 105.$$210÷2=105.

Step 2: Divide $105$105 by $3$3:

$$105 div 3 = 35.$$105÷3=35.

Step 3: Divide $35$35 by $5$5:

$$35 div 5 = 7.$$35÷5=7.

Step 4: $7$7 is prime, so stop.

Final Prime Factorization:

$$210 = 2 times 3 times 5 times 7.$$210=2×3×5×7.

20. A number $n$n has a prime factorization of $2^3 times 3^2 times 7$23×32×7. What is the number, and is it composite?

Step 1: Calculate the number $n$n:

$$n = 2^3 times 3^2 times 7 = 8 times 9 times 7 = 72 times 7 = 504.$$n=23×32×7=8×9×7=72×7=504.

Step 2: Determine if composite:
Since $504$504 has multiple prime factors, it is composite.

21. Which of the following numbers are prime: $29, 45, 67, 91$29,45,67,91?

Step 1: Test divisibility of each number:

• $29$29: Divisible only by $1$1 and $29$29 → Prime.
• $45$45: Divisible by $3, 5$3,5 → Composite.
• $67$67: Divisible only by $1$1 and $67$67 → Prime.
• $91$91: Divisible by $7$7 → Composite.

Answer: Prime numbers are $29$29 and $67$67.

22. Write the prime factorization of $360$360.

Step 1: Divide $360$360 by $2$2 repeatedly:

$$360 div 2 = 180, quad 180 div 2 = 90, quad 90 div 2 = 45.$$360÷2=180,180÷2=90,90÷2=45.

Step 2: Divide $45$45 by $3$3:

$$45 div 3 = 15, quad 15 div 3 = 5.$$45÷3=15,15÷3=5.

Step 3: $5$5 is prime.

Final Prime Factorization:

$$360 = 2^3 times 3^2 times 5.$$360=23×32×5.

23. Find the smallest number divisible by $2, 3, 5, text{and } 7$2,3,5,and 7.

Step 1: Find the LCM of $2, 3, 5, text{and } 7$2,3,5,and 7:

$$text{LCM} = 2 times 3 times 5 times 7 = 210.$$LCM=2×3×5×7=210.

Answer: $210$210.

24. A number has only two factors, $1$1 and $23$23. Is it prime or composite?

Step 1: Definition of prime number:
A prime number has only two factors: $1$1 and itself.

Step 2: Check $23$23:
$23$23 has exactly two factors ($1$1 and $23$23).

Answer: $23$23 is prime.

25. A number has a prime factorization of $2^2 times 3 times 11$22×3×11. How many total factors does it have?

Step 1: Formula for total factors:
If $n = p^a times q^b times r^c$n=pa×qb×rc, then the total number of factors is:

$$(a+1)(b+1)(c+1).$$(a+1)(b+1)(c+1).

Step 2: Substitute values:

$$n = 2^2 times 3 times 11 quad Rightarrow quad (2+1)(1+1)(1+1) = 3 times 2 times 2 = 12.$$n=22×3×11⇒(2+1)(1+1)(1+1)=3×2×2=12.

Answer: The number has $12$12 total factors.