Divisibility and Multiples | FIO

Question 29

Choose any 3 numbers. When is their sum divisible by 3? Explore all possible cases and generalise.

Solution

Understand the Question

Any whole number can be expressed in the form $3k + r$ , where $r \in \{0, 1, 2\}$ is the remainder when divided by $3$ .
The sum of three numbers $A + B + C$ is divisible by $3$ if and only if the sum of their remainders $(r_1 + r_2 + r_3)$ is divisible by $3$ .
We explore all possible remainder combinations to find every case where their sum is a multiple of $3$ , then generalise.

Step 1 · Represent Numbers in Terms of Remainders

Let the three numbers be $A, B,$ and $C$ . Each number can be written in terms of its quotient and remainder when divided by $3$ :

\begin{aligned} A &= 3k_1 + r_1 \\ B &= 3k_2 + r_2 \\ C &= 3k_3 + r_3 \end{aligned}

where $k_1, k_2, k_3$ are whole numbers and each remainder $r_i \in \{0, 1, 2\}$ . Diagram 1

Sum of the three numbers:

\begin{aligned} A + B + C &= (3k_1 + r_1) + (3k_2 + r_2) + (3k_3 + r_3) \\ &= 3(k_1 + k_2 + k_3) + (r_1 + r_2 + r_3) \end{aligned}

Since $3(k_1 + k_2 + k_3)$ is always divisible by $3$ , the sum $A + B + C$ is divisible by $3$ if and only if $(r_1 + r_2 + r_3)$ is divisible by $3$ .

Step 2 · Find Possible Sums of Remainders

Since each remainder $r_i \in \{0, 1, 2\}$ :

Minimum sum of remainders $= 0 + 0 + 0 = 0$
Maximum sum of remainders $= 2 + 2 + 2 = 6$

The possible values of $(r_1 + r_2 + r_3)$ that are divisible by $3$ are $0$ , $3$ , and $6$ .

Step 3 · Explore Cases for Divisibility

Case 1: Sum of remainders is $0$

This occurs when $r_1 = 0, r_2 = 0, r_3 = 0$ (all three numbers are divisible by $3$ ).

Example: $3, 6, 9$

\begin{aligned} 3 \div 3 &\implies \text{remainder } 0 \\ 6 \div 3 &\implies \text{remainder } 0 \\ 9 \div 3 &\implies \text{remainder } 0 \end{aligned}

$\text{Sum of remainders} = 0 + 0 + 0 = 0$ $\text{Sum of numbers} = 3 + 6 + 9 = 18 \quad (\text{divisible by } 3)$

Case 2: Sum of remainders is $3$

This occurs in two ways:

(a) One remainder of each type ( $0, 1, 2$ ):

Example: $3, 4, 5$

\begin{aligned} 3 \div 3 &\implies \text{remainder } 0 \\ 4 \div 3 &\implies \text{remainder } 1 \\ 5 \div 3 &\implies \text{remainder } 2 \end{aligned}

$\text{Sum of remainders} = 0 + 1 + 2 = 3$ $\text{Sum of numbers} = 3 + 4 + 5 = 12 \quad (\text{divisible by } 3)$

(b) All three remainders are $1$ :

Example: $1, 4, 7$

\begin{aligned} 1 \div 3 &\implies \text{remainder } 1 \\ 4 \div 3 &\implies \text{remainder } 1 \\ 7 \div 3 &\implies \text{remainder } 1 \end{aligned}

$\text{Sum of remainders} = 1 + 1 + 1 = 3$ $\text{Sum of numbers} = 1 + 4 + 7 = 12 \quad (\text{divisible by } 3)$

Case 3: Sum of remainders is $6$

This occurs when $r_1 = 2, r_2 = 2, r_3 = 2$ (all three remainders are $2$ ).

Example: $2, 5, 8$

\begin{aligned} 2 \div 3 &\implies \text{remainder } 2 \\ 5 \div 3 &\implies \text{remainder } 2 \\ 8 \div 3 &\implies \text{remainder } 2 \end{aligned}

$\text{Sum of remainders} = 2 + 2 + 2 = 6$ $\text{Sum of numbers} = 2 + 5 + 8 = 15 \quad (\text{divisible by } 3)$

Step 4 · Generalise the Conditions

Combining all successful cases, the sum of three numbers is divisible by $3$ in exactly two scenarios:

All three numbers have the same remainder when divided by $3$ (all remainders $0$ , all $1$ , or all $2$ ).
The three numbers have all distinct remainders when divided by $3$ (one remainder $0$ , one $1$ , and one $2$ ).

Answer

The sum of any $3$ numbers is divisible by $3$ if and only if:

(i) All three numbers leave the same remainder when divided by $3$ (all $0$ , all $1$ , or all $2$ ).

(ii) All three numbers leave different remainders when divided by $3$ (one $0$ , one $1$ , and one $2$ ).

Common Mistakes

Assuming all numbers must be multiples of 3: Forgetting that numbers with remainders like $1+1+1$ (e.g. $1+4+7=12$ ) or $2+2+2$ (e.g. $2+5+8=15$ ) also add up to a multiple of $3$ .
Missing the mixed case: Overlooking the combination $0 + 1 + 2 = 3$ , where picking one number of each remainder type (e.g. $3 + 4 + 5 = 12$ ) yields a sum divisible by $3$ .
Two-same, one-different error: Incorrectly assuming two numbers with remainder $1$ and one with remainder $0$ could work ( $1+1+0=2$ , not divisible by $3$ ).

More questions in FIO

The sum of four consecutive numbers is 34. What are these numbers?

Suppose $p$ is the greatest of five consecutive numbers. Describe the other four numbers in terms of $p$ .

For each statement below, determine whether it is always true, sometimes true, or never true. Explain your answer. Mention examples and non-examples as appropriate. Justify your claim using algebra.

(i) The sum of two even numbers is a multiple of 3.

(ii) If a number is not divisible by 18, then it is also not divisible by 9.

(iii) If two numbers are not divisible by 6, then their sum is not divisible by 6.

(iv) The sum of a multiple of 6 and a multiple of 9 is a multiple of 3.

(v) The sum of a multiple of 6 and a multiple of 3 is a multiple of 9.

Find a few numbers that leave a remainder of 2 when divided by 3 and a remainder of 2 when divided by 4. Write an algebraic expression to describe all such numbers.

“I hold some pebbles, not too many, When I group them in 3’s, one stays with me. Try pairing them up — it simply won’t do, A stubborn odd pebble remains in my view. Group them by 5, yet one’s still around, But grouping by seven, perfection is found. More than one hundred would be far too bold, Can you tell me the number of pebbles I hold?”

Tathagat has written several numbers that leave a remainder of 2 when divided by 6. He claims, “If you add any three such numbers, the sum will always be a multiple of 6.” Is Tathagat’s claim true?

When divided by 7, the number 661 leaves a remainder of 3, and 4779 leaves a remainder of 5. Without calculating, can you say what remainders the following expressions will leave when divided by 7? Show the solution both algebraically and visually.

(i) $4779 + 661$

(ii) $4779 - 661$

Find a number that leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and a remainder of 4 when divided by 5. What is the smallest such number? Can you give a simple explanation of why it is the smallest?

Find, without dividing, whether the following numbers are divisible by 9.

(i) 123 (ii) 405 (iii) 8888 (iv) 93547 (v) 358095

Q10

Find the smallest multiple of 9 with no odd digits.

Q11

Find the multiple of 9 that is closest to the number 6000.

Q12

How many multiples of 9 are there between the numbers 4300 and 4400?

Q13

The digital root of an 8-digit number is 5. What will be the digital root of 10 more than that number?

Q14

Write any number. Generate a sequence of numbers by repeatedly adding 11. What would be the digital roots of this sequence of numbers? Share your observations.

Q15

What will be the digital root of the number $9a + 36b + 13$ ?

Q16

Make conjectures by examining if there are any patterns or relations between (i) the parity of a number and its digital root. (ii) the digital root of a number and the remainder obtained when the number is divided by 3 or 9.

Q17

If $31z5$ is a multiple of $9$ , where $z$ is a digit, what is the value of $z$ ? Explain why there are two answers to this problem.

Q18

"I take a number that leaves a remainder of 8 when divided by 12. I take another number which is 4 short of a multiple of 12. Their sum will always be a multiple of 8", claims Snehal. Examine his claim and justify your conclusion.

Q19

When is the sum of two multiples of 3, a multiple of 6 and when is it not? Explain the different possible cases, and generalise the pattern.

Q20

Sreelatha says, "I have a number that is divisible by 9. If I reverse its digits, it will still be divisible by 9".

(i) Examine if her conjecture is true for any multiple of 9.

(ii) Are any other digit shuffles possible such that the number formed is still a multiple of 9?

Q21

If $48a23b$ is a multiple of $18$ , list all possible pairs of values for $a$ and $b$ .

Q22

If $3p7q8$ is divisible by 44, list all possible pairs of values for $p$ and $q$ .

Q23

Find three consecutive numbers such that the first number is a multiple of 2, the second number is a multiple of 3, and the third number is a multiple of 4.

Are there more such numbers? How often do they occur?

Q24

Write five multiples of 36 between 45,000 and 47,000. Share your approach with the class.

Q25

The middle number in the sequence of 5 consecutive even numbers is $5p$ . Express the other four numbers in sequence in terms of $p$ .

Q26

Write a 6-digit number that it is divisible by 15, such that when the digits are reversed, it is divisible by 6.

Q27

Deepak claims, "There are some multiples of 11 which, when doubled, are still multiples of 11. But other multiples of 11 don't remain multiples of 11 when doubled". Examine if his conjecture is true; explain your conclusion.

Q28

Determine whether the statements below are 'Always True', 'Sometimes True', or 'Never True'. Explain your reasoning.

(i) The product of a multiple of 6 and a multiple of 3 is a multiple of 9.

(ii) The sum of three consecutive even numbers will be divisible by 6.

(iii) If $abcdef$ is a multiple of 6, then $badcef$ will be a multiple of 6.

(iv) $8(7b - 3) - 4(11b + 1)$ is a multiple of 12.

Q29

Choose any 3 numbers. When is their sum divisible by 3? Explore all possible cases and generalise.

Q30

Is the product of two consecutive integers always multiple of 2? Why? What about the product of these consecutive integers? Is it always a multiple of 6? Why or why not? What can you say about the product of 4 consecutive integers? What about the product of five consecutive integers?

Q31

Solve the cryptarithms —

(i) $\text{EF} \times \text{E} = \text{GGG}$

(ii) $\text{WOW} \times 5 = \text{MEOW}$

Q32

Which of the following Venn diagrams captures the relationship between the multiples of 4, 8, and 32?