June 30th, 2014, 10:05 PM  #1 
Member Joined: Jun 2014 From: Brighton Posts: 49 Thanks: 2  Find out the least number
Help me to solve math homework problem. What is the least number, which when divided by 12, 15, 20 and 54 leaves remainder of 8 in each case? Please tell me the procedure to solve this problem. Last edited by skipjack; July 1st, 2014 at 01:54 AM. 
July 1st, 2014, 01:56 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,057 Thanks: 1618 
Obviously, the least natural number with that property is 8.

July 1st, 2014, 02:51 AM  #3 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 
If the trivial solution 8 is not allowed or not good then the answer is $\displaystyle LCM (12,15,20,54) + 8$ Where LCM is the lowest common multiple. This could be calculated like this : $\displaystyle \frac {12*15*20*54 } { GCD(12,15,20,54) } + 8$ Where GCD is the greatest common divisor. The greatest common divisor is obviously 1. And thus the final answer is $\displaystyle (12*15*20*54) + 8 \; = \; 194408$ Last edited by gelatine1; July 1st, 2014 at 03:43 AM. 
July 1st, 2014, 01:30 PM  #4 
Senior Member Joined: Apr 2014 From: Europa Posts: 571 Thanks: 176 
$\displaystyle \rm \ L \ C\ M\ (12,\ 15,\ 20,\ 54) = 540$

July 1st, 2014, 03:47 PM  #5 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 
Whoops then I assume I extended the formula $\displaystyle LCM(a,b) = \frac {a*b}{GCD(a,b)}$ in a wrong way... The result could have been calculated as follows: $\displaystyle LCM(12,15,20,54)+8 = LCM \left( \frac {12*15}{GCD(12,15)} , \frac {20*54}{GCD(20,54)} \right) + 8$ $\displaystyle = LCM (60,540) + 8 = \frac {60*540}{GCD(60,540)} + 8 = 548$ Which should be correct this time. My apologies for the wrong answer before. 
July 1st, 2014, 04:38 PM  #6 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,606 Thanks: 845  
July 2nd, 2014, 03:08 AM  #7 
Newbie Joined: Jun 2014 From: United state Posts: 14 Thanks: 0 Math Focus: algebra, geometry, trigonometry, calculus 
Find the LCM of 12,15,20,54, The LCM will 6,15,10,27 Thus we will find the ans. The ans will be 548 For more details you can take help online tutor. 
July 2nd, 2014, 04:34 AM  #8 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,606 Thanks: 845  
July 2nd, 2014, 04:35 AM  #9 
Senior Member Joined: Apr 2014 From: Europa Posts: 571 Thanks: 176 
$\displaystyle \Large{\color{blue}{\ \qquad 12=2^2\cdot3 \\\;\\ \ \qquad 15=3\cdot5 \\\;\\ \ \qquad 20=2^2\cdot5 \\\;\\ \ \qquad 54=2\cdot3^3 \\ \_\_\_\_\_\_\_\_\_\_\_\_\_ \\\;\\ LCM = 2^2\cdot3^3\cdot5 = 4\cdot27\cdot5 = 20\cdot27 = 540 .}} $ 
July 2nd, 2014, 04:39 AM  #10 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11  In case the numbers are very big it will be faster to calculate the gcd (using the euclidean algorithm) instead of factorising the numbers.


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