Consecutive sums
Every odd number splits in two
An odd number is one more than twice something, so split it into halves differing by one.
📚 Chapter Learning Kit
Number Play · Class 8 Mathematics · Ganita Prakash
English 3 topics 105 leveled MCQs 51 flashcards 5 games Free
Shared by a Veda teacher · Generated with Veda AI · Oct 2026
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More than one way
Some numbers have several forms
Twenty-one is 10 plus 11, and also 6 plus 7 plus 8, and also 1 up to 6.
Powers of two
The numbers that refuse
Try to write 8, 16 or 32 as a sum of consecutive numbers and you will fail.
Two even numbers added
Sometimes yes, sometimes no
Twenty plus 36 is 56, a multiple of 4. Twenty plus 14 is 34, which is not.
Sorted by the remainder on 4
Even numbers come in two kinds
Divide an even number by 4 and the remainder is 0 or 2. It cannot be 1 or 3.
The first case
Two multiples of 4
Add 4p and 4q and take the 4 out: it is 4 times the quantity p plus q.
📖 Smart notes
What you'll study, topic by topic
1
Parity, and answers you can predict
Put plus and minus signs between four numbers in all eight ways and every answer has the same parity. A short piece of algebra shows why, and the same reasoning tells you which expressions must come out even.
- Every odd number splits into two whole numbers in a row.
- Some numbers can be written as consecutive sums in several ways.
- A power of two cannot be written as a sum of consecutive numbers.
~35 min · full explanation, examples & memory tricks in the app
2
What divides a sum
If a number divides two others, it divides their sum and their difference. The reverse is not safe. Sorting even numbers by the remainder they leave on 4 settles when two of them add to a multiple of 4.
- Even numbers leave a remainder of either 0 or 2 on division by 4.
- Two multiples of 4 add to a multiple of 4.
- Two evens that each leave 2 also add to a multiple of 4.
~35 min · full explanation, examples & memory tricks in the app
3
Why the divisibility shortcuts work
Write a number by its place values and every shortcut explains itself. Each power of ten is one more than a multiple of nine. That is the whole secret of the digit sum, and a near miss of eleven gives that rule too.
- Any number is the sum of its place values.
- Every place value above the units is a multiple of 10.
- So only the last digit decides divisibility by 10, 5 and 2.
~35 min · full explanation, examples & memory tricks in the app
❓ Leveled MCQ practice
Try the smart MCQs from this kit
105 questions laddered from warm-up to topper-level, each with an explanation. A taste:
Anshu writes 7 as 3 plus 4 and 12 as 3 plus 4 plus 5. Each of these is an example of a number written as
Beginner
A a product of two factors
B a difference of two squares
C a multiple of its digit sum
D a sum of consecutive numbers
Show answer & explanation
a sum of consecutive numbers
The numbers added run one after another with no gaps.
Split 9 into two whole numbers that follow one another. They are
Beginner
A 3 and 6
B 2 and 7
C 4 and 5
D 4 and 6
Show answer & explanation
4 and 5
Nine is one more than twice four, so the halves differ by one.
EVERY odd number splits into two whole numbers in a row. The reason is that an odd number is
Advanced
A always a multiple of three
B one more than twice something
C never a multiple of two
D the sum of its own digits
Show answer & explanation
one more than twice something
That extra one is what lets the two halves differ by one.
Of these forms, the one that does NOT add up to 21 is
Advanced
A 4 plus 5 plus 6 plus 7
B 10 plus 11
C 6 plus 7 plus 8
D 1 plus 2 plus 3 plus 4 plus 5 plus 6
Show answer & explanation
4 plus 5 plus 6 plus 7
Those four numbers come to 22, while the other three forms all give 21.
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