Digits & Number Properties

Formula sheet

Everything for this chapter, in one place.

Place value — the one tool

Reversal


Two digits, reversed minus original	$9(b-a)$
Three digits, reversed minus original	$99(c-a)$ — the middle digit never matters
So a reversal difference	is always a multiple of 9, or of 99
A palindrome	reads the same both ways

Digit sums and divisibility


Divisible by 3 or 9	the digit sum is
Divisible by 11	the alternating digit sum is
Divisible by 4	the last two digits form a multiple of 4
Divisible by 8	the last three digits do
Divisible by 2, 5, 10	decided by the last digit alone

Big powers, written out


$2^a \times 5^b$ with $a>b$	$= 2^{a-b} \times 10^{b}$ — a short number followed by $b$ zeros
$10^n - k$	a run of 9s followed by the digits of $10^m - k$
So a digit sum	comes from the shape of the number, never from multiplying it out

Counting with digit conditions


All digits distinct, 3-digit	$9 \times 9 \times 8$
Product of digits is non-zero	no digit may be 0
Mean of a symmetric family	the mean of each digit position, recombined

Magic square, $3\times3$ with 1–9


Every line sums to	15
The centre	is always 5

Why this chapter matters

0.5

questions in a typical Quant section

20th

of 25 chapters

≈ 1 of 66 marks a section ride on it.

Asked in10 of 24 papers

20172025

Year by yearper section

Question type12 asked

MCQ 42%Type-in 58%

Type-in answers carry no negative marking.

From every Digits & Number Properties question in CAT papers since 2017.