1. Express $21$ as the sum of two consecutive natural numbers.
Answer:
Let the numbers be $x$ and $x+1$.
$$x+(x+1)=21$$
$$2x+1=21$$
$$x=10$$
Therefore, the numbers are 10 and 11.
2. Write $18$ in two different ways as a sum of consecutive natural numbers.
Answer:
$$3+4+5+6=18$$
$$5+6+7=18$$
Thus, two representations are $3+4+5+6$ and $5+6+7$.
3. Can an even number be expressed as the sum of two consecutive natural numbers? Give a reason.
Answer: No. Two consecutive natural numbers consist of one even and one odd number. Their sum is always odd. Therefore, their sum cannot be an even number.
4. Explain why the sum of two consecutive natural numbers is always odd.
Answer: If the consecutive numbers are $n$ and $n+1$, then
$$n+(n+1)=2n+1$$
Since $2n$ is even, $2n+1$ is always odd.
5. Find all possible ways of expressing $15$ as a sum of consecutive natural numbers.
Answer:
$$7+8=15$$
$$4+5+6=15$$
$$1+2+3+4+5=15$$
Therefore, the possible representations are $7+8$, $4+5+6$, and $1+2+3+4+5$.
6. Three consecutive natural numbers have a sum of $51$. Find the numbers.
Answer:
Let them be $x$, $x+1$, $x+2$.
$$x+(x+1)+(x+2)=51$$
$$3x+3=51$$
$$3x=48$$
$$x=16$$
Hence, the numbers are 16, 17 and 18.
7. Three consecutive numbers add up to $72$. Find the numbers.
Answer:
$$x+(x+1)+(x+2)=72$$
$$3x+3=72$$
$$3x=69$$
$$x=23$$
Therefore, the numbers are 23, 24 and 25.
8. If three consecutive natural numbers have a total of $96$, what are the numbers?
Answer:
$$3x+3=96$$
$$3x=93$$
$$x=31$$
Therefore, the numbers are 31, 32 and 33.
9. A student claims that $10$ can be written as the sum of two consecutive natural numbers. Is the claim correct?
Answer: No. If the numbers are $x$ and $x+1$,
$$2x+1=10$$
$$2x=9$$
$$x=4.5$$
Since $x$ is not a natural number, 10 cannot be expressed as the sum of two consecutive natural numbers.
11. Explain, with an example, why a number ending in 0 is divisible by 10.
Answer: A number ending in $0$ has $10$ as a factor. For example,
$$350\div10=35$$
Hence, 350 is divisible by 10.
12. Without performing long division, find the remainder when $583$ is divided by $9$.
Answer: Add the digits:
$$5+8+3=16$$
$$1+6=7$$
Therefore, the remainder is 7.
13. Use the divisibility rule of 9 to determine whether $4572$ is divisible by 9.
Answer:
$$4+5+7+2=18$$
Since $18$ is divisible by $9$, 4572 is divisible by 9.
14. What should be done with the digits of a number to test its divisibility by 9?
Answer: Add all the digits of the number. If the resulting sum is divisible by $9$, the original number is also divisible by $9$.
15. State the divisibility rule for 2 and give two examples.
Answer: A number is divisible by $2$ if its units digit is 0, 2, 4, 6 or 8.
Examples: 246 and 580.
16. Which of the following are divisible by 5: $235$, $348$, $570$, $821$?
Answer: 235 and 570 are divisible by 5 because their units digits are $5$ and $0$, respectively.
17. What condition must the last two digits of a number satisfy for the number to be divisible by 4?
Answer: The number formed by the last two digits must be divisible by $4$.
18. How is the divisibility rule for 8 different from that for 4?
Answer:
- For divisibility by $4$, we check the last two digits.
- For divisibility by $8$, we check the last three digits.
19. A number is divisible by both 2 and 3. What can we conclude about its divisibility by 6?
Answer: The number is divisible by 6, because a number divisible by both $2$ and $3$ is divisible by their product, $6$.
20. Two consecutive natural numbers have a sum of $47$. Find them. What does this tell us about the sum of consecutive numbers?
Answer:
$$x+(x+1)=47$$
$$2x+1=47$$
$$x=23$$
Therefore, the numbers are 23 and 24. This illustrates that the sum of two consecutive natural numbers is always odd.
21. Find the remainder when $764$ is divided by $9$ without using long division.
Answer: Add the digits:
$$7+6+4=17$$
$$1+7=8$$
Therefore, the remainder is 8.
22. Ravi wants to express $35$ as the sum of two consecutive natural numbers. Find the numbers.
Answer:
$$x+(x+1)=35$$
$$2x+1=35$$
$$x=17$$
Therefore,
$$35=17+18$$
23. Three consecutive natural numbers have a sum of $117$. Find the numbers.
Answer:
$$x+(x+1)+(x+2)=117$$
$$3x+3=117$$
$$3x=114$$
$$x=38$$
Hence, the numbers are 38, 39 and 40.
24. A number has a digit sum of $18$. What can you say about its divisibility by 9?
Answer: Since $18$ is divisible by $9$, the original number is divisible by 9.
25. A student says, “If the digit sum of a number is divisible by 3, the number must be divisible by 9.” Is this always true? Give a counterexample.
Answer: No. For example, $12$ has digit sum
$$1+2=3$$
so it is divisible by $3$, but $12$ is not divisible by 9.
26. A number has 7 in its units place. Can it be divisible by 5? Explain.
Answer: No. A number divisible by $5$ must have 0 or 5 in its units place. Since the units digit is $7$, it cannot be divisible by $5$.
27. A student checks only the last digit to determine whether $3248$ is divisible by 4. Explain why this is insufficient.
Answer: For divisibility by $4$, we must examine the last two digits, not just the last digit. Here the last two digits are $48$, and
$$48\div4=12$$
Therefore, 3248 is divisible by 4.
28. Consider the number $7216$. Determine whether it is divisible by 2, 4, 8 and 9 without performing long division.
Answer:
- By $2$: last digit is $6$ → Yes
- By $4$: last two digits are $16$ → Yes
- By $8$: last three digits are $216$ and $216\div8=27$ → Yes
- By $9$: $7+2+1+6=16$ → No
Therefore, 7216 is divisible by 2, 4 and 8, but not by 9.
1.
Which of the following numbers is divisible by $9$?
A. 4,326
B. 5,214
C. 7,418
D. 6,125
Answer: B. 5,214
2.
A number has digits whose sum is $27$. Which conclusion is necessarily correct?
A. The number is divisible only by $3$
B. The number is divisible by $9$
C. The number is divisible by $5$
D. The number is divisible by $11$
Answer: B. The number is divisible by $9
3.
If a number is divisible by $12$, which of the following must also divide it?
A. 5
B. 7
C. 6
D. 11
Answer: C. 6
4.
If a number is divisible by $24$, which pair of numbers must both divide it?
A. 3 and 8
B. 5 and 6
C. 7 and 12
D. 4 and 5
Answer: A. 3 and 8
5.
What is the digital root of $57,428$?
A. 8
B. 7
C. 6
D. 5
Answer: A. 8
Explanation:
$$5+7+4+2+8=26$$
$$2+6=8$$
6.
The digital root of a number is obtained by repeatedly:
A. Multiplying its digits
B. Subtracting its digits
C. Adding its digits until a single digit remains
D. Dividing the number by $9$
Answer: C. Adding its digits until a single digit remains
7.
What is the digital root of $6,237$?
A. 3
B. 6
C. 9
D. 0
Answer: C. 9
8.
Which statement about a positive multiple of $9$ is correct?
A. Its digital root is always $3$
B. Its digital root is always $6$
C. Its digital root is always $9$
D. Its digital root can never be determined
Answer: C. Its digital root is always $9
9.
Which statement is always true?
A. The sum of two multiples of $8$ is divisible by $8$
B. The sum of two odd numbers is odd
C. The sum of an odd and an even number is even
D. Every multiple of $6$ is odd
Answer: A. The sum of two multiples of $8$ is divisible by $8
10.
A number is divisible by $9$ if:
A. Its last digit is $9$
B. Its first digit is $9$
C. The sum of its digits is divisible by $9$
D. Its last two digits are divisible by $9$
Answer: C. The sum of its digits is divisible by $9$
11.
The number $42x7$ is divisible by $9$. What can be the value of $x$?
A. 1
B. 2
C. 5
D. 6
Answer: C. 5
Explanation:
$$4+2+x+7=13+x$$
For divisibility by $9$:
$$13+x=18$$
Therefore,
$$x=5$$
12.
What is the result when an odd number is added to an even number?
A. Always even
B. Always odd
C. Always a multiple of 4
D. Sometimes even and sometimes odd
Answer: B. Always odd
13.
Which method can be used to test the divisibility of a number by $11$?
A. Add all the digits
B. Check the last digit
C. Find the alternating sum of the digits
D. Check the last three digits
Answer: C. Find the alternating sum of the digits
14.
Consider the number $572$. Using the alternating-sum method, which expression should be evaluated?
A. $5+7+2$
B. $5-7+2$
C. $5+7-2$
D. $7-5-2$
Answer: B. $5-7+2$
15.
Using the divisibility rule for $11$, is $121$ divisible by $11$?
A. Yes, because $1-2+1=0$
B. No, because the digit sum is $4$
C. Yes, because its last digit is $1$
D. No, because it is an odd number
Answer: A. Yes, because $1-2+1=0$
16.
Which of the following can never be the result of adding an odd number and an even number?
A. 13
B. 21
C. 34
D. 47
Answer: C. 34
17.
Which statement is correct?
A. Every odd number is a multiple of $6$
B. Every multiple of $6$ is even
C. Every even number is a multiple of $6$
D. Every multiple of $3$ is a multiple of $6$
Answer: B. Every multiple of $6$ is even
18.
The product of two consecutive integers is always divisible by:
A. 2
B. 3
C. 5
D. 6
Answer: A. 2
19.
If two consecutive integers are represented by $n$ and $n+1$, which expression represents their product?
A. $2n+1$
B. $n^2+1$
C. $n(n+1)$
D. $2n$
Answer: C. $n(n+1)$
20.
Which pair demonstrates that the product of two consecutive integers is divisible by $2$?
A. $3$ and $5$
B. $6$ and $7$
C. $8$ and $10$
D. $4$ and $6$
Answer: B. $6$ and $7$
21.
The product of four consecutive integers is always divisible by:
A. 4
B. 6
C. 12
D. 24
Answer: D. 24
24.
What is the digital root of $31$?
A. 3
B. 4
C. 5
D. 6
Answer: B. 4