Identity

Place-value expansion of a numeral

Arithmetic • Number System • Digit Problems

Formula

ab c = 100 a + 10 b + c

A three-digit numeral with digits

a, b, c

has value

ab c = 100 a + 10 b + c

, where

a \in {1, \dots, 9}

and

b, c \in {0, \dots, 9}

; the same pattern extends to any number of digits.

How to use it

1Expand every numeral in a digit puzzle, then collect the known place values on one side.
2Reversing the digits gives $100 c + 10 b + a$ , so the difference is $99 (a - c)$ .
3Carry the digit ranges along: they are the constraints that pin the answer down.

A two-digit number is

4

times the sum of its digits. Find all such numbers.

Step-by-step

1
$10 a + b = 4 (a + b)$
Why: Expand the numeral by place value.
2
$6 a = 3 b \Rightarrow b = 2 a$
3
$a = 1, 2, 3, 4 \Rightarrow 12, 24, 36, 48$
Why: $b = 2 a \leq 9$ caps $a$ at $4$ .

Answer

12, 24, 36, 48

Watch out

The leading digit is never

0

. And

ab c

denotes the numeral, never the product

a \cdot b \cdot c

Also known as

digits to number, 10a+b form, expanded form of a number