Divisibility and Multiples | FIO

Question 3

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.

Solution

Understand the Question

To determine whether a general mathematical statement is always true, sometimes true, or never true:

Always true: Must hold for all possible integers. We prove this algebraically using general variable representations (such as $2a$ for even numbers or $3k$ for multiples of 3).
Sometimes true: Holds for some specific numbers (examples) but fails for others (non-examples/counterexamples).
Never true: Fails for every single case, or the logical implication is consistently invalidated by counterexamples.

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

Step 1 · Analyze Algebraically and Test with Examples

Let the two even numbers be $2a$ and $2b$ , where $a$ and $b$ are whole numbers.

2a + 2b = 2(a + b)

For $2(a + b)$ to be a multiple of $3$ , $(a + b)$ must be divisible by $3$ , which is not true for all pairs of numbers.

Example (True case): Let the numbers be $6$ ( $a = 3$ ) and $12$ ( $b = 6$ ). $\text{Sum} = 6 + 12 = 18 = 3 \times 6$ $18$ is a multiple of $3$ .
Non-example (False case): Let the numbers be $4$ ( $a = 2$ ) and $10$ ( $b = 5$ ). $\text{Sum} = 4 + 10 = 14$ $14$ is not a multiple of $3$ .

Since the statement holds for some cases but fails for others, it is sometimes true.

Answer

(i) Sometimes true

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

Step 1 · Analyze Using Divisibility and Algebra

A conditional statement "If $P$ , then $Q$ " is false if we find a number where $P$ is true (not divisible by $18$ ) but $Q$ is false (divisible by $9$ ).

Let a number divisible by $9$ be $N = 9k$ :

If $k$ is even ( $k = 2m$ ), $N = 9(2m) = 18m$ , which is divisible by $18$ .
If $k$ is odd ( $k = 2m + 1$ ), $N = 9(2m + 1) = 18m + 9$ .

Here, $N = 18m + 9$ is divisible by $9$ , but leaves a remainder of $9$ when divided by $18$ (not divisible by $18$ ).

Counterexamples: Numbers like $9, 27, 45$ are not divisible by $18$ , yet they are divisible by $9$ .

Since any odd multiple of $9$ violates the claim, the statement is never true as a general rule.

Answer

(ii) Never true

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

Step 1 · Test with Examples and Counterexamples

We test pairs of numbers that are not divisible by $6$ :

Example (True case): Let the numbers be $1$ and $2$ . $\text{Sum} = 1 + 2 = 3$ Neither $1$ nor $2$ is divisible by $6$ , and their sum $3$ is also not divisible by $6$ .
Counterexample (False case): Let the numbers be $4$ and $2$ . $\text{Sum} = 4 + 2 = 6$ Neither $4$ nor $2$ is divisible by $6$ , but their sum $6$ is divisible by $6$ .

Since it holds in some cases and fails in others, the statement is sometimes true.

Answer

(iii) Sometimes true

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

Step 1 · Prove Algebraically

Let a multiple of $6$ be $6a$ and a multiple of $9$ be $9b$ , where $a$ and $b$ are whole numbers.

6a + 9b = 3(2a + 3b)

Since $a$ and $b$ are whole numbers, $(2a + 3b)$ is also a whole number. Therefore, $6a + 9b$ is always divisible by $3$ .

Examples:
- $12 + 18 = 30 = 3 \times 10$
- $6 + 45 = 51 = 3 \times 17$

The statement is always true.

Answer

(iv) Always true

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

Step 1 · Analyze Algebraically and Test with Examples

Let a multiple of $6$ be $6a$ and a multiple of $3$ be $3b$ , where $a$ and $b$ are whole numbers.

6a + 3b = 3(2a + b)

For $3(2a + b)$ to be a multiple of $9$ , $(2a + b)$ must be divisible by $3$ , which does not hold for all whole numbers $a$ and $b$ .

Example (True case): For $a = 1, b = 1$ : $\text{Sum} = 6(1) + 3(1) = 6 + 3 = 9$ $9$ is a multiple of $9$ .
Non-example (False case): For $a = 1, b = 2$ : $\text{Sum} = 6(1) + 3(2) = 6 + 6 = 12$ $12$ is not a multiple of $9$ .

Since the statement holds for some cases but not all, it is sometimes true.

Answer

(v) Sometimes true

Common Mistakes

Assuming a single example proves a rule: Showing a statement works for one pair of numbers (e.g. $6+3=9$ ) does not make it always true. Always test algebraic general forms.
Confusing factors and multiples in divisibility: If a number is divisible by $18$ , it must be divisible by $9$ (since $9$ is a factor of $18$ ). However, the reverse is not true: numbers like $9$ and $27$ are divisible by $9$ but not by $18$ .

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?