Question

I. Find the lowest number which is increased by 3 is exactly divisible by 8, 12 and 16.
II. Find the lowest number which is less by 5 is exactly divisible by 15, 25 and 50.

Answer 1

Hint: In both the questions given above we can observe one thing that the answer should be common for the three numbers. Like the number is divisible by 8, 12 and 16 and in other cases it should be divisible by 15, 25 and 50. This is the hint that it should be the LCM on which the remaining condition of increased or decreased number should be applied. So we will find the LCM first and then will operate the conditions as per the question.

Complete step by step solution:
Case I:
Given the numbers are 8, 12 and 16.
Now to find the LCM we will take the multiples of these numbers from which we will select the lowest common multiple.
Multiples of 8 are 8, 16, 24, 32, 40 , 48, 56….
Multiples of 12 are 12, 24, 36, 48, 60, 72…
Multiples of 16 are 16, 32, 48, 64, 80,…
Now we can observe that the Lowest common multiple is 48.
Thus this is the number which is divisible by 8, 12 and 16 exactly.
But the condition is that the number that is divisible is when it is increased by 3. Thus we will subtract 3 from the LCM above to get the exact answer.
Thus \[48 - 3 = 45\] is the number we need.
Because if we add 3 to this number we will get 48 and it is divisible by 8, 12 and 16.
Thus the answer is 45.

Case II:
Given the numbers are 15, 25 and 50.
Now to find the LCM we will use the prime factorization method. This will take the help of prime numbers to write the given number in product form.
\[
  15 = 5 \times 3 \\
  25 = 5 \times 5 \\
  50 = 5 \times 5 \times 2 \;
\]
Now LCM will be the product of common and uncommon factors.
So LCM is \[5 \times 5 \times 3 \times 2 = 150\]
But we know that the condition is the number when less by 5 will be divisible by 15, 25 and 50.
Thus we will add 5 to this number.
So the number is \[150 + 5 = 155\] is the number we need.
So that when we remove 5 from 155 we get 150 and it is divisible by 15, 25 and 50.
Thus the answer is 155.

Note: Note that we are not finding the lowest number that can divide the given numbers instead we are finding the number that is divisible by these three numbers given in the separate questions. Both the conditions are totally different.
Also note that when we find LCM ; that is not our answer rather we are nearer to the answer. Such that we will find the number nearer to it by either adding or removing the numbers given in the condition to satisfy the question.
The methods used to find LCM in both cases are used differently so that a student can take help of that which is easier for him/her.