determine the smallest four digit number that has exactly divisible by 8 , 10 and 14
Answers
Answer:
I’ll go through a step by step, as detailed as possible path on solving this problem, and the algorithm can be applied in any similar case.
Let’s do prime factorization for the three numbers to find the least common multiple (LCM) for the given numbers:
8 = 2 * 2 * 2
10 = 2 * 5
12 = 2 * 2 * 3
As far as 3 and 5 are odd numbers, we’ll have the following to construct the LCM for the three numbers:
2*2*2, the number’s divisible by 8
2*2*2*5 will make the number divisible by 10 (LCM for 8 and 10)
2*2*2*5*3 the number’s now divisible by 12 (LCM for 8, 10, 12)
So we have 2*2*2*3*5 = 120 and it’s the LCM of 8, 10 and 12.
The smallest four digit number is 1000, right? :)
So now we have to find the smallest four digit number that is a multiple of 120, let’s go:
a)
1000 / 120 = 8.33
8 < 8.333 < 9
8 * 120 = 960 (it is not a four digit number)
9 * 120 = 1080 (or 960 +120) which is a four digit number and it’s the smallest four digit number which is divisible by 8, 10 and 12.
OR
b)
1000 / 120 = 8 (remainder = 40)
1000 + 120 - 40 = 1080
Thus, 1080 is the least common four digit multiple of 8, 10 and 12 (smallest 4 digit number that is exactly divisible by 8, 10 and 12 like you have stated in the question).
1120
Step-by-step explanation:
LCM of 8, 10 and 14 is 2×4×5×7 = 280
Smallest four digit multiple of 280 is
280×4=1120