Divisibility and Multiples | FIO

Question 18

"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.

Solution

We will represent the two numbers using algebraic expressions based on the given conditions and then find their sum to check for divisibility by 8.

Step 1 — Representing the numbers

Let the first number be $N_1$ . It leaves a remainder of 8 when divided by 12. So, we can write $N_1$ as $12k + 8$ , where $k$ is any integer.

Let the second number be $N_2$ . It is 4 short of a multiple of 12. So, we can write $N_2$ as $12m - 4$ , where $m$ is any integer.

Step 2 — Finding their sum

Let $S$ be the sum of these two numbers. We add the expressions for $N_1$ and $N_2$ .

$S = N_1 + N_2$

$S = (12k + 8) + (12m - 4)$

$S = 12k + 12m + 8 - 4$

$S = 12k + 12m + 4$

$S = 12(k + m) + 4$

Let $K = k + m$ . Since $k$ and $m$ are integers, $K$ is also an integer. So, the sum can be written as:

$\boxed{S = 12K + 4}$

Step 3 — Examining divisibility by 8

For Snehal's claim to be true, the sum $S$ must always be a multiple of 8. This means $S$ should be expressible as $8 \times (\text{some integer})$ .

We have $S = 12K + 4$ . We can factor out 4 from this expression.

$S = 4(3K + 1)$

For $S$ to be a multiple of 8, $4(3K + 1)$ must be a multiple of 8. This means the term $(3K + 1)$ must be an even number. Let us check if $(3K + 1)$ is always even.

Case 1: If $K$ is an even integer. Let $K = 2p$ for some integer $p$ . Then $3K + 1 = 3(2p) + 1 = 6p + 1$ . This expression $6p + 1$ is always an odd number. So, if $K$ is even, $S = 4 \times (\text{an odd number})$ . This means $S$ is a multiple of 4, but not a multiple of 8.

Case 2: If $K$ is an odd integer. Let $K = 2p + 1$ for some integer $p$ . Then $3K + 1 = 3(2p + 1) + 1 = 6p + 3 + 1 = 6p + 4$ . This expression $6p + 4$ is always an even number. So, if $K$ is odd, $S = 4 \times (\text{an even number})$ . This means $S$ is a multiple of 8.

Since $K = k+m$ can be either even or odd, the sum $S$ is not always a multiple of 8. Snehal's claim is false.

Step 4 — Providing a counterexample

Let us choose specific values for $k$ and $m$ such that $K = k+m$ is an even number. Let $k=0$ and $m=0$ . Then $K = 0+0 = 0$ , which is an even number.

The first number $N_1 = 12k + 8$ . $N_1 = 12(0) + 8$ $N_1 = 8$ The second number $N_2 = 12m - 4$ . $N_2 = 12(0) - 4$ $N_2 = -4$ The sum $S = N_1 + N_2$ . $S = 8 + (-4)$

$\boxed{S = 4}$ The number 4 is not a multiple of 8. This example shows that the sum is not always a multiple of 8.

Answer

Snehal's claim is false. The sum of such numbers is not always a multiple of 8. For example, if the first number is 8 (which leaves a remainder of 8 when divided by 12) and the second number is -4 (which is 4 short of a multiple of 12), their sum is 4, which is not a multiple of 8.

More questions in FIO

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

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Q10

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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

Q19

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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

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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

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Q28

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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

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Q32

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