Number Play

Why This Matters

Numbers are not just things you count with. They have secret habits.

Some numbers can be broken into a neat ladder of consecutive numbers, like 15 = 4 + 5 + 6. Some numbers refuse to be broken that way, no matter how hard you try. Some numbers can be checked for “is it divisible by 9?” in a few seconds, just by adding their digits — without doing any division at all.

These look like magic tricks. But they are not magic. Every one of them has a clear reason behind it. Once you see the reason, the trick becomes obvious, and you can never forget it.

This chapter is about playing with numbers and asking “why?” about everything you notice. Why can every odd number be split into two consecutive numbers? Why does adding the digits tell you about division by 9? Why do some patterns always hold?

By the end, you will not just know a few shortcuts. You will understand why they work. And understanding why is the real power of mathematics — it lets you predict patterns you have never even seen before, instead of memorising hundreds of separate rules.

The Big Idea

Numbers are built from simple parts, and those parts follow simple rules. An even number always has a 2 hidden inside it. A multiple of 9 always has a “9 + 1” hidden in every place value. Once you write a number using its hidden parts — using a bit of algebra or a picture of dots — every pattern explains itself. Instead of memorising what happens, you can prove what must happen, for every number at once, without checking them one by one.

Let’s Break It Down

We will explore four big ideas. First, which numbers are sums of consecutive numbers, and why. Then parity (whether a number is odd or even) and how it behaves. Then quick divisibility tests for 9, 3 and 11, and the reasons they work. Finally, patterns in factors, multiples and digital roots.

Before we start, let us quickly refresh a few words we will use again and again.

Sums of consecutive numbers

Anshu noticed something fun. Many numbers can be written as a sum of consecutive numbers:

7 = 3 + 4, 12 = 3 + 4 + 5, 15 = 7 + 8 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5.

Some numbers can even be split in more than one way, like 15 above. Let us see this with dots, because a picture makes the idea click.

The diagram below shows 15 split three different ways. Each time, the same 15 dots are just regrouped.

Tap to zoom

Now for the big question: can every number be written this way? And which numbers can be split in many ways?

Start with the easiest pattern. Every odd number is the sum of two consecutive numbers. For example, 13 = 6 + 7, and 25 = 12 + 13. Why is this always true?

Here is the reason, shown with dots. The picture below splits 13 into two near-equal rows.

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Let us make that reasoning exact with a tiny bit of algebra.

What about other numbers? It turns out almost every number can be written as a sum of consecutive numbers — except the powers of 2: 1, 2, 4, 8, 16, 32, and so on. These are the only numbers that simply refuse. (The deep reason is studied in higher classes, but you can test it yourself: try to split 8 or 16 into consecutive numbers and you will fail every time.)

Parity — how odd and even behave

Here is a surprising pattern from the textbook. Take any 4 consecutive numbers, say 3, 4, 5, 6. Put + or − signs between them in every possible way:

3 + 4 + 5 + 6 = 18, 3 + 4 − 5 + 6 = 8, 3 − 4 − 5 − 6 = −12, and so on.

There are 8 such expressions. If you work them all out, you notice something: the answer is always even. Never odd. And this happens no matter which 4 numbers you start with. Why?

The secret is parity. Let us first nail down how odd and even behave when you add or subtract.

odd ± odd = even

even ± even = even

odd ± even = odd

Notice the most important fact here: changing a + into a − never changes the parity. For example, 7 + 3 = 10 (even) and 7 − 3 = 4 (also even). The parity stays the same.

Here is the clean reason, straight from the textbook. Take an expression like a + b − c − d. Now flip one sign, say change + b to − b, to get a − b − c − d. How much did the value change?

(a + b − c − d) − (a − b − c − d) = 2b

The change is 2b, which is always even. So flipping a sign moves the value by an even amount. And if two numbers differ by an even amount, they must have the same parity. So flipping signs never changes whether the answer is odd or even.

Starting from one expression, you can reach all 8 by flipping signs. So all 8 expressions have the same parity. For 3, 4, 5, 6 the plain sum 3 + 4 + 5 + 6 = 18 is even, so all 8 must be even too.

Let us use this idea to judge an expression without computing it.

Be careful though — not every expression is always even. Take x² + 2. If x is even, x² is even, so x² + 2 is even. But if x is odd, x² is odd, so x² + 2 is odd. For example, x = 3 gives 9 + 2 = 11, which is odd. So x² + 2 is only sometimes even.

When do two even numbers add to a multiple of 4?

Take two even numbers and add them. Is the sum always a multiple of 4? Try a few:

2 + 6 = 8 (yes, multiple of 4), but 2 + 4 = 6 (no, not a multiple of 4).

So sometimes yes, sometimes no. When exactly? The trick is to sort even numbers into two types, by their remainder when divided by 4.

Type A — multiples of 4: numbers like 4, 8, 12, 16. They leave remainder 0 when divided by 4. In algebra: 4p.

Type B — even but not multiples of 4: numbers like 2, 6, 10, 14. They leave remainder 2 when divided by 4. In algebra: 4q + 2.

Now there are three cases. Two Type-A numbers, two Type-B numbers, or one of each. Let us picture the trickiest one: two Type-B numbers, like 6 + 10. The diagram below arranges every number in rows of 4 and shows what happens to the leftovers.

Tap to zoom

In algebra this is just as clean: (4p + 2) + (4q + 2) = 4p + 4q + 4 = 4(p + q + 1), which is a multiple of 4. The two 2s combine into a 4.

Here are all three cases together. Notice the pattern is exactly like the parity of odd and even.

What you add	Algebra	Is the sum a multiple of 4?
multiple of 4 + multiple of 4	4p + 4q = 4(p + q)	Yes — always
remainder-2 + remainder-2	(4p + 2) + (4q + 2) = 4(p + q + 1)	Yes — the two 2s make a 4
multiple of 4 + remainder-2	4p + (4q + 2) = 4(p + q) + 2	No — remainder 2 is left over

So two even numbers add to a multiple of 4 exactly when they are both Type A, or both Type B. A mismatch leaves a remainder of 2.

Always, sometimes, or never — divisibility facts

The same kind of reasoning settles many divisibility statements. The big tools are these four facts, all provable with the “factor out” trick:

If a divides M and a divides N, then a divides M + N and M − N.

If A is divisible by k, then every multiple of A is divisible by k.

If A is divisible by k, then A is divisible by every factor of k.

If A is divisible by k and by m, then A is divisible by the LCM of k and m.

The first one is easy to see. If 8 divides two numbers, say 8a and 8b, then their sum is 8a + 8b = 8(a + b), which is still a multiple of 8. The shared factor of 8 never disappears when you add.

Let us use these to judge a statement.

A quick test for divisibility by 9 (and why it works)

Here is a famous shortcut: a number is divisible by 9 if the sum of its digits is divisible by 9. For example, 5 + 4 = 9 tells you 54 is divisible by 9. No division needed.

But why does just adding digits work? This is the heart of the chapter. The reason is beautiful, and the picture below shows it using 427.

Tap to zoom

So the rule is really about remainders. Every place value (10, 100, 1000, …) is one more than a multiple of 9. The multiple-of-9 parts contribute nothing to the remainder. What is left over is exactly the digit sum. If the digit sum is a multiple of 9, the whole number is too.

Let us use it.

The test for 3 works exactly the same way, for the same reason: every place value is also one more than a multiple of 3 (10 = 9 + 1, 100 = 99 + 1, and 9 and 99 are multiples of 3 too). So a number is divisible by 3 if the sum of its digits is divisible by 3.

A quick test for divisibility by 11 (and why it works)

The test for 11 has a twist: you alternately add and subtract the digits. Why the alternating signs? Because the place values do not all behave the same way — they flip between one more and one less than a multiple of 11. The picture below shows this.

Tap to zoom

So the rule is: start at the units digit, alternately add and subtract the digits, and if the result is 0 or a multiple of 11, the number is divisible by 11. Let us try it.

Factors, multiples and digital roots

Many number patterns come from how factors nest inside one another. A clear example: the multiples of 4, 8 and 32. Every multiple of 32 is also a multiple of 8, and every multiple of 8 is also a multiple of 4. The picture below shows this as circles inside circles.

Tap to zoom

This is just the third tool from before: if a number is divisible by 32, it is divisible by every factor of 32, including 8 and 4.

Now a fun pattern called the digital root. Take a number, add its digits, then keep adding the digits of the result until a single digit is left. That single digit is the digital root.

For example, for 489710: 4 + 8 + 9 + 7 + 1 + 0 = 29, then 2 + 9 = 11, then 1 + 1 = 2. So the digital root of 489710 is 2.

This is the very same process as the divisibility test for 9. So the digital root tells you the remainder when the number is divided by 9. (If the digital root is 9, the number is an exact multiple of 9.)

Common Mistakes

These are the slip-ups students make most often in this chapter. Spot them now to keep your reasoning clean.

Quick Check

Try these quick questions. The explanation appears after you answer, so read it either way.

Practice Problems

Try each one yourself first. Then tap to check your full solution.

Easy

Medium

Challenge

Summary

• A run of consecutive numbers is a stack of growing rows. Every odd number is the sum of two consecutive numbers (2n + 1 = n + (n + 1)). Every number except a power of 2 (1, 2, 4, 8, 16, …) can be written as a sum of consecutive numbers.
• Parity (odd/even) follows simple rules: odd ± odd = even, even ± even = even, odd ± even = odd. Flipping a + to a − changes the value by an even amount, so it never changes the parity.
• An expression is always even if you can factor out a 2, like 4m + 2q = 2(2m + q).
• Two even numbers add to a multiple of 4 exactly when both are multiples of 4, or both leave remainder 2. A mismatch leaves remainder 2.
• A number is divisible by 9 (or 3) if its digit sum is, because every place value is one more than a multiple of 9 (and of 3).
• A number is divisible by 11 if the alternating sum of its digits is 0 or a multiple of 11, because place values flip between one more and one less than a multiple of 11.
• If a number is divisible by k, it is divisible by every factor of k; every multiple of it keeps the same divisors; and if divisible by k and m, it is divisible by their LCM.
• A digital root (repeatedly adding digits to one digit) equals the remainder on dividing by 9.

What’s Next

You have just seen the real power of mathematics: not memorising rules, but proving why they must be true — with a picture of dots or a line of algebra.

Next, in Chapter 6 — We Distribute, Yet Things Multiply, you will meet the distributive law and see how breaking numbers apart and putting them back together unlocks fast multiplication and powerful algebra. The “factor it out” trick you used all through this chapter is exactly the idea that comes next. Onward!

Go to Chapter 6 — We Distribute, Yet Things Multiply →