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Question

Find the greatest number of 4 digits which is exactly divisible by 15 24 and 36.


Solution

Hey,

The answer is : 9720

Here is a step - by - step procedure to find the greatest four digit number that is exactly divisible by 15,24,36.

  1. Find the LCM (Least common multiple) of 15, 24, 36. Any number divisible by the LCM of the 15,24,36 will be divisible by each of 15,24,36.
  • To find LCM, write each number as a product of its prime factors.
  • 15=3∗5=3∗5 ———————————————15 has one 3 and one 5.
  • 24=2∗2∗2∗3 ———————————24 has three 2’s and one 3.
  • 36=2∗2∗3∗3 ———————————36 has two 2’s and two 3’s.
  • To get the LCM: Multiply each factor the greatest number of times it occurs in any of the numbers.
  • There are 3 factors : 2,3,5
  • The greatest number of times 2 occurs in the numbers (15, 24, 36) : Three
  • The greatest number of times 3 occurs in the numbers (15, 24, 36) : Two
  • The greatest number of times 5 occurs in the numbers (15, 24, 36) : One
  • LCM =2∗2∗2∗3∗3∗5=360

2. To Find the greatest four digit number divisible by 360:

  • The greatest four digit number is 9999
  • 9999 when divided by 360 is 27.75 ( 27.75 is not an integer, thus 9999 is not divisible by 360. )
  • The greatest four digit number divided by 360 would be = 360∗27=9720

( Note : 27 is the part of the number before the decimal point in 27.75)

3. We can check if 9720 is divisible by each of 15, 24, 36

  • 9720/15=648
  • 9720/24=405
  • 9720/36=270

Mathematics

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