Try this beautiful Positive Integer Problem from Algebra from PRMO 2017, Question 1.

## Positive Integer – PRMO 2017, Question 1

How many positive integers less than 1000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by $3 ?$

- $9$
- $7$
- $28$

**Key Concepts**

Algebra

Equation

multiplication

## Check the Answer

But try the problem first…

Answer:$28$

PRMO-2017, Problem 1

Pre College Mathematics

## Try with Hints

First hint

Let $n$ be the positive integer less than 1000 and $s$ be the sum of its digits, then $3 \mid n$ and $7 \mid s$

$3|n \Rightarrow 3| s$

therefore$21| s$

Can you now finish the problem ……….

Second Hint

Also $n<1000 \Rightarrow s \leq 27$

therefore $\mathrm{s}=21$

Clearly, n must be a 3 digit number Let $x_{1}, x_{2}, x_{3}$ be the digits, then $x_{1}+x_{2}+x_{3}=21$

where $1 \leq x_{1} \leq 9,0 \leq x_{2}, x_{3} \leq 9$

$\Rightarrow x_{2}+x_{3}=21-x_{1} \leq 18$

$\Rightarrow x_{1} \geq 3$

Can you finish the problem……..

Final Step

For $x_{1}=3,4, \ldots ., 9,$ the equation (1) has $1,2,3, \ldots ., 7$ solutions

therefore total possible solution of equation (1)

=$1+2+\ldots+7=\frac{7 \times 8}{2}=28$

## Other useful links

- https://www.cheenta.com/ordered-pairs-prmo-2019-problem-18/
- https://www.youtube.com/watch?v=h_x9kS-J1XY

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