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# (i) If $34x$ is a multiple of $3$, where $x$ is a digit, what is the value of $x$?(ii) If $74x5284$ is a multiple of $3$, where $x$ is a digit, find the value(s) of $x$.

Last updated date: 15th Aug 2024
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Hint:
Here we are given two multiples of three. We are asked to find the unknown digit. For that we can use the idea that if a number is a multiple of three, then the sum of its digits are also multiples of three.

Complete step by step solution:
(i) It is given that $34x$ is a multiple of three.
We have to find the digit $x$.
We know if a number is a multiple of three, then the sum of its digits is also a multiple of three. So we have,
$3 + 4 + x$ is a multiple of three.
$\Rightarrow 7 + x$ is a multiple of $3$.
We have the smallest multiple of three greater than $7$ and $9$.
This gives, $7 + x = 9$
So we get, $x = 9 - 7 = 2$
Therefore $x = 2$ is a possibility.
$7 + x$ is a multiple of $3$ and also gives the possibilities $7 + x = 12,7 + x = 15$.
This gives, $x = 12 - 7 = 5.x = 15 - 7 = 8$.
So $x = 5,x = 8$ are also possibilities.
But if $7 + x = 18$ then $x = 18 - 7 = 11$. This is not possible since $x$ is a single digit.
So we have three possible values for $x$; $x = 2,5,8$.

(ii) It is given that $74x5284$ is a multiple of three.
So here too we can say that the sum of the digits is a multiple of three.
This gives, $7 + 4 + x + 5 + 2 + 8 + 4$ is a multiple of three.
$\Rightarrow 30 + x$ is a multiple of $3$.
Since $30$ itself is a multiple of three, we can see that $x$ is also a multiple of three (since the difference of two multiples of $3$ is again a multiple of $3$).
So possibilities of $x$ are $0, 3, 6, 9$.

Note:
So here we used the idea that if a number is a multiple of three, the sum of the digits are also multiples of three. The point should be careful is that the unknown digit may have more than one possibility. So do not stop after getting one favourable answer.