INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

February 20, 2020

Divisibility - Understanding the Rule : AMC 8, 2016 Problem 24

What is Divisibility?

Divisibility is the study of finding remainder when a number is divided by another number.

Try the problem

The digits $1$, $2$, $3$, $4$, and $5$ are each used once to write a five-digit number $PQRST$. The three-digit number $PQR$ is divisible by $4$, the three-digit number $QRS$ is divisible by $5$, and the three-digit number $RST$ is divisible by $3$. What is $P$?

$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$

AMC 8, 2016 Problem number 24


6 out of 10

Mathematical Circles

Knowledge Graph

Divisibility Rule- Knowledge graph

Use some hints

See this image:

Divisibility rule step 1

So we have $5$ blank space to fill with the numbers $1,2,3,4,5$ following the given restriction.

See the given conditions care fully :

$1.\quad PQR$ is divisible by $4$,

$2.\quad QRS$ is divisible by $5$,

$3. \quad RST$ is divisible by $3$

Can you make a definite choice for any of the places ???

$1.\quad PQR$ is divisible by $4 \Rightarrow$ $QR$ is divisible by $4$

$2.\quad QRS$ is divisible by $5 \Rightarrow $, $S$ is either $5$ or $0$. [But we don't have $0$ so $S$ must be $5$]

$3. \quad RST$ is divisible by $3 \Rightarrow $ $R+S+T$ is divisible by 3.

Divisibility Rule- Step 2

Try to find out other choices, other blanks !!

$QR$ is divisible by $4$ so $R$ needs to be $2$ or $4$.

Let $R=2$

$\Rightarrow R+S+T=2+5+T=7+T$

As, $R+S+T$ is divisible by 3 then possible values of $T$ are $2, 5, 8$

We have already used $2$ and $5$. And $8$ is not in the given set of numbers. (Contradiction)

Therefore $R \ne 2 \Rightarrow R = 4$

Divisibility Rule - Step 3

Now the things are easy, try to drive it from here !!!!!!!!!

Again, $RST$ is divisible by $3$

$\Rightarrow R+S+T$ is divisible by $3$

$\Rightarrow 4+5+T$ is divisible by $3$

Then the possibilities for $T$ are $3, 6$

$6$ is not in the given number, hence $T=3$

 Step 4

We have two numbers left and two blank places left.

if $Q=1$

then $QR=14$ is not divisible by $4$

Then $Q=2$ [$24$ is divisible by $4$]

And certainly $P=1$

The number is :

 Step 5

Subscribe to Cheenta at Youtube

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.