CAT 2025 Lesson : Divisibility - 7, 13 and Composite

2.5 Divisor is $7$ or $13$

Rule: To find the remainder when a number is divided by $7$ or $13$, from the sum of alternating set of $3$ digits starting from the right (hundreds' , tens' and units' places), subtract the sum of remaining set of $3$ digits and then divide by $7$ or $13$.

Explanation

$7 \times 11 \times 13 = 1001$

$\text{Rem} \left(\dfrac{1000^0}{7}\right) ; \text{Rem} \left(\dfrac{1000^1}{7} \right) = -1 ;$

$\text{Rem} \left(\dfrac{1000^2}{7} \right)= 1 ; \text{Rem} \left(\dfrac{1000^3}{7} \right) = -1$

As noted above, $1$ and $-1$ are the remainders when even and odd powers of $1000$ are divided by $7$ respectively. The same applies when the divisor is $13$.

∴ $\text{Rem} \left(\dfrac{1000^n}{7 \space \text{or} \space 13} \right)= 1$ if $n$ is even and (-1) if $n$ is odd.

Note: Questions pertaining to divisibility rule of $7$ or $13$ are very rare.

2.6 Divisor is any composite number

Divisibility Rule: Prime factorise the number and check for divisibility by each of the prime factors raised to their respective powers.

Remainder Rule: To find the remainder, apply Chinese Remainder theorem (covered in the Factors & Remainders lesson).

If we take $24 (=2^{3} \times 3)$, we need to check for divisibility by $2^{3}$ and $3$.

Likewise, to check for divisibility by $84 (=2^2 \times 3 \times 7)$, we need to check for divisibility by $2^{2}, 3$ and $7$.

2.7 Summary of Divisibility Rules

Below is a summary of divisibility rules, which are derived from the rules for remainders. Please remember these for ease in calculation.

Number	Rule
$2$	Last digit divisible by $2$
$3$	Sum of digits divisible by $3$
$4$	Last two digits divisible by $4$
$5$	Last digit is $0$ or $5$
$6$	Divisible by $2$ and $3$
$7$	Difference between sum of alternate sets of digits is divisible by $7$
$8$	Last three digits divisible by $8$
$9$	Sum of digits divisible by $9$
$10$	Last digit is $0$
$11$	Difference between sum of alternate sets of digits is divisible by $11$
$12$	Divisible by $3$ and $4$
$13$	Difference between sum of alternate sets of digits is divisible by $13$
$14$	Divisible by $2$ and $7$
$15$	Divisible by $3$ and $5$
$16$	Last $4$ digits divisible by $16$
$18$	Divisible by $2$ and $9$
$20$	Divisible by $4$ and $5$
$24$	Divisible by $3$ and $8$
$25$	Last $2$ digits divisble by $25$
$32$	Last $5$ digits divisible by $32$
$100$	Last $2$ digits are $0$
$125$	Last $3$ digits divisible by $125$