Question

A heap of stones can be made up into groups of 21. When made up into groups 16, 20, 25 and 45 there are three stones left in each case. How many stones at least can there be in the heap?
(a)7203
(b)2403
(c)3603
(d)4803

Answer 1

Hint: It is given that heap of stones can be made up into groups of 21, meaning it is divisible by 21. And for groups 16, 20, 25 and 45 there are three stones left in each case means the remainder is three. We will check all the options one by one and see which option satisfies all the conditions and that will be the answer.

Complete step-by-step answer:
Let’s first check option (a),
Let’s see if 7203 is divisible by 21 or not,
$7203=21\times 343$
Hence, 1st condition is satisfied.
Now if 7203 gives remainder 3 then 7200 must be divisible by 16, 20, 25 and 45.
Let’s check one by one,
$\begin{align}
  & 7200=16\times 450 \\
 & 7200=20\times 360 \\
 & 7200=25\times 288 \\
 & 7200=45\times 160 \\
\end{align}$
Hence, option (a) can be correct.
Let’s check option (b),
Let’s see if 2403 is divisible by 21 or not,
$2403=21\times 114+9$
Hence, it is not divisible by 21.
Therefore, option (b) is incorrect.
Let’s check option (c),
Let’s see if 3603 is divisible by 21 or not,
$3603=21\times 171+12$
Hence, it is not divisible by 21.
Hence, option (c) is incorrect.
Let’s check option (d),
Let’s see if 4803 is divisible by 21 or not,
$4803=21\times 228+15$
Hence, it is not divisible by 21.
Hence, option (d) is incorrect.
So, we get only option (a) which satisfies all the conditions.
Hence, option (a) is correct.

Note: One can also check the divisibility of 7 and 3 separately instead of 21 directly so that it will become easy to check that it is divisible by 21 or not. Once we have checked one condition and if it doesn’t satisfy that number then there is no need to check another condition as it will consume more time.