Divisibility and Multiples

Anshu is exploring sums of consecutive numbers. He has written the following:

Now, he is wondering—

• "Can I write every natural number as a sum of consecutive numbers?"
• "Which numbers can I write as the sum of consecutive numbers in more than one way?"
• "Ohh, I know all odd numbers can be written as a sum of two consecutive numbers. Can we write all even numbers as a sum of consecutive numbers?"
• "Can I write 0 as a sum of consecutive numbers? Maybe I should use negative numbers."

Explore these questions and any others that may occur to you. Discuss them with the class.

Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place '+' and '-' signs in between the numbers. How many different possibilities exist? Write all of them.

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

Find the smallest multiple of 9 with no odd digits.

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

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

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

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.

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

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.

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.

"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.

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.

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?

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

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

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?

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

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

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

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.

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.

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

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?

Solve the cryptarithms —

(i) $\text{EF} \times \text{E} = \text{GGG}$ (ii) $\text{WOW} \times 5 = \text{MEOW}$

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

Evaluate each expression and write the result next to it. Do you notice anything interesting?

Now, take four other consecutive numbers. Place the '+' and '-' signs as you have done before. Find out the results of each expression. What do you observe?

Repeat this for one more set of 4 consecutive numbers. Share your findings.

Do these patterns occur no matter which 4 consecutive numbers are chosen? Is there a way to find out through reasoning?

Hint: Use algebra and describe the 8 expressions in a general form.

Now take any 4 numbers, place '+' and '-' signs in the eight different ways, and evaluate the resulting expression. What do you observe about their parities?

Repeat this with other sets of 4 numbers.

Is there a way to explain why this happens?

Hint: Think of the rules for parity of the sum or difference of two numbers.

Context: Now, let us see what happens when a negative sign is switched to a positive sign.

Q. Replace any negative sign in the expression $a + b - c - d$ with a positive sign and find the difference between the two numbers.

Context: Replace any negative sign in the expression $a + b - c - d$ with a positive sign and find the difference between the two numbers.

Q. What do you conclude from this observation?

Is the phenomenon of all the expressions having the same parity limited to taking 4 numbers? What do you think?

Breaking Even

We know how to identify even numbers. Without computing them, find out which of the following arithmetic expressions are even.

Using our understanding of how parity behaves under different operations, identify which of the following algebraic expressions give an even number for any integer values for the letter-numbers.

Context: The algebraic expressions from the previous page are:

Q. Similarly, determine and explain which of the other expressions always give even numbers. Write a couple of examples and non-examples, as appropriate, for each expression.

Write a few algebraic expressions which always give an even number.

Pairs to Make Fours

Take a pair of even numbers. Add them. Is the sum divisible by 4?

Try this with different pairs of even numbers. When is the sum a multiple of 4, and when is it not? Is there a general rule or a pattern?

When will two even numbers add up to give a multiple of 4?

This problem is similar to the question of identifying when adding two numbers will result in an even number. Can you see this?

There are three cases to examine:

Look at the following expressions and the visualisation. Write the corresponding explanation and examples.

Always, Sometimes, or Never

We examine different statements about factors and multiples and determine whether a statement is 'Always True', 'Sometimes True', or 'Never True'.

We know that the sum of any two multiples of 2 is also a multiple of 2.

1. If 8 exactly divides two numbers separately, it must exactly divide their sum.

Statement 1 is always true. Determine if it is true with subtraction.

Examine each of the following statements, and determine whether it is 'Always true', 'Sometimes true', 'Never true'.

6. If a number is divisible by both 9 and 4, it must be divisible by 36.

7. If a number is divisible by both 6 and 4, it must be divisible by 24.

Context: Let us consider another expression, $5k - 2$, and see the values it takes for different values of $k$.

Numbers that leave a remainder of $3$ when divided by $5$ can also be seen as $2$ less than multiples of $5$; $5k - 2$, where $k \ge 1$.

Q. Are there other expressions that generate numbers that are $3$ more than a multiple of $5$?

Similarly, explain using algebra why the divisibility shortcuts for 5, 2, 4, and 8 work.

Look at each of the following statements. Which are correct and why?

(i) If a number is divisible by 9, then the sum of its digits is divisible by 9.

(ii) If the sum of the digits of a number is divisible by 9, then the number is divisible by 9.

(iii) If a number is not divisible by 9, then the sum of its digits is not divisible by 9.

(iv) If the sum of the digits of a number is not divisible by 9, then the number is not divisible by 9.

The shortcut to find the divisibility by 3 is similar to the method for 9. A number is divisible by 3 if the sum of its digits is divisible by 3. Explore the remainders when powers of 10 are divided by 3. Explain why this method works.

Using these observations, can you tell whether the number 462 is divisible by 11?

Context: This alternating pattern of one more than 11 and one less than 11 continues for higher place values. Since 400 contains 4 hundreds, 400 is 4 more than a multiple of 11 ($396 + 4$). Since 60 contains 6 tens, 60 is 6 less than a multiple of 11 ($66 - 6$). Since 2 contains 2 units, 2 is 2 more than a multiple of 11, i.e., $2 = (0 + 2)$. Using these observations, can you tell whether the number 462 is divisible by 11?

Q. What could be a general method or shortcut to check divisibility by 11?

If this difference is 11 or a multiple of 11, what does that say about the remainder obtained when the number is divisible by 11?

Using this shortcut, find out whether the following numbers are divisible by 11. Further, find the remainder if the number is not divisible by 11.

(i) 158 (ii) 841 (iii) 481 (iv) 5529 (v) 90904 (vi) 857076

Is this method similar to or different from the method we saw just before?

Fill in the following table. Find a quick way to do this?

More on Divisibility Shortcuts

Divisibility Shortcuts for Other Numbers

How can we find out if a number is divisible by 6?

Will checking its divisibility by its factors 2 and 3 work? Use the shortcuts for 2 and 3 on these numbers and divide each number by 6 to verify— 38, 225, 186, 64.

How about checking divisibility by 24? Will checking the divisibility by its factors, 4 and 6, work? Why or why not?

What property do you think this digital root will have? Recall that we did this while finding the divisibility shortcut for 9.

Between the numbers 600 and 700, which numbers have the digital root: (i) 5, (ii) 7, (iii) 3?

Write the digital roots of any 12 consecutive numbers. What do you observe?

Now, find the digital roots of some consecutive multiples of (i) 3, (ii) 4, and (iii) 6.

What are the digital roots of numbers that are 1 more than a multiple of 6? What do you notice?

Try to explain the patterns noticed.

I’m made of digits, each tiniest and odd, No shared ground with root #1—how odd!

My digits count, their sum, my root— All point to one bold number’s pursuit— The largest odd single-digit I proudly claim.

What’s my number? What’s my name?

Solve the cryptarithms given below:

Try this now: GH × H = 9K.

This means a 2-digit number multiplied by a 1-digit number gives another 2-digit number in the 90s. Observe the letters corresponding to the units digits in this cryptarithm. Pick the solution to this question from the options given below: 11 × 9 = 99, 12 × 8 = 96, 46 × 2 = 92, 24 × 4 = 96, 47 × 2 = 94, 31 × 3 = 93, 16 × 6 = 96.

Solve the following:

(i) UT × 3 = PUT

(ii) AB × 5 = BC

(iii) L2N × 2 = 2NP

(iv) XY × 4 = ZX

(v) PP × QQ = PRP

(vi) JK × 6 = KKK