Tips on Cracking Aptitude Questions Based on LCM & HCF

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5 TIPS on cracking Aptitude Questions on LCM HCF
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Tip 1 Remember the basic formulae

• Least Common Multiple or LCM of A and B is the
  smallest number that can be divided by A or B
  without any reminder.
• Highest Common Factor or HCF (or GCD Greatest
  Common Divisor) of A and B is the largest number
  that can divide A or B without any reminder.
• While determining the LCM or HCF, express the
  terms as products of prime numbers. For instance,
  let the terms be 15, 25 and 27. We can factorize
  these terms as
• 15 5 x 3
• 25 52
• 27 33
• HCF 1 no prime number is found across all
  three terms
• LCM 52 x 33 we take the highest power for each
  prime number from the factorization results
• Product of 2 numbers LCM x HCF This only
  holds for 2 terms

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Tip 2 Steps to calculate the LCM and HCF of
fractions

• HCF of fractions HCF of Numerator/ LCM of
  Denominator
• Proof
• Let the fractions be a/b and c/d. Let the HCF be
  x/y.
• The HCF is the highest possible number that
  divides both fractions without a reminder. Since
  we are looking for the highest possible number,
  we must try to maximize x and minimize y.
• a/b y/x should have no reminder.
• c/d y/x should have no reminder.
• gtx is the HCF of a and c to ensure that x
  divides a and c both and is as large as possible
• gty is the LCM of b and d to ensure that y is
  divided by b and d both and is as small as
  possible.
• LCM of fractions LCM of Numerator/ HCF of
  denominator
• Proof
• Let the fractions be a/b and c/d. Let the LCM be
  x/y
• The LCM is the smallest possible number that can
  be divided by both fractions without a reminder.
  Since we are looking for the smallest possible
  number, we must try to minimize x and maximize y.
• x/y b/a should have no reminder.
• x/y d/c should have no reminder.
• gtx is the LCM of a and c to ensure that x is
  divided by both a and c and is as small as
  possible
• gty is the HCF of b and d to ensure that y
  divides both b and d and is as large as possible

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Tip 3 Understand the concepts clearly to figure
where to use HCF and where to use LCM

• Question Find the greatest 4-digit number which
  is divisible by 15, 25, 40 and 75.
• Solution LCM of 15, 25, 40, 75 600. We need to
  find the largest 4 digit number divisible by 600.
• Largest 4-digit number 9999. Remainder on
  dividing 9999 by 600 399.
• Thus, largest 4-digit number divisible by 15, 25,
  40 and 75 9999 399 9600.
• Question The ratio of two numbers is 3 4 and
  their H.C.F. is 4. Find their LCM.
• Solution Let the numbers be 3x and 4x. HCF of 3
  and 4 is 1. Therefore, HCF of 3x and 4x x
• x 4.
• Thus, the numbers are 12 and 16. LCM 48
• Question A, B and C start at the same time in
  the same direction to run around a circular
  stadium. A completes a round in 252 seconds, B in
  308 seconds and C in 198 seconds. After what time
  will they all meet again at the starting point,
  if they keep running in circles?
• Solution We need to find the time after which
  each of A, B and C wouldve completed full
  rounds.
• LCM of 252, 308, 198 2272 sec 2272 / 60
  minutes 46 min 12 sec.
• Question Six bells commence tolling together and
  toll at intervals of 2, 4, 6, 8 10 and 12 seconds
  respectively. In 30 minutes, how many times do
  they toll together?

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Tip 4 The smallest number which when divided by
x, y and z leaves the same remainder R in each
case LCM(x, y, z) R
Question Find the smallest number which when
divided by 5, 6 , 7 and 8 leaves a remainder 3,
but when divided by 9 leaves no
remainder. Solution The number must (i) be a
multiple of 9 and (ii) leave a remainder of 3
when divided by 5, 6, 7 and 8. LCM of 5, 6, 7
and 8 840. Adding the remainder, the required
number (840 x t) 3. Now, smallest value of t
for which (840 x t) 3 is divisible by 9 2
Trial and error Thus, the required number is
(840 x 2) 3 1683.
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Tip 5 Largest number which divides x, y, z (x gt
y, y gt z) to leave same remainder HCF(x y, y
z, x z)
In case that the remainder is same and given, say
R, the required number will be the HCF of x R,
y R, z R. Similarly, the largest number
which divides x, y and z to leave remainders a, b
and c respectively is the HCF of x a, y b, z
c. Question Find the greatest number that
will divide 183, 91 and 43 so as to leave the
same remainder in each case. Solution Let x be
the greatest possible number such that it leaves
the same reminder when it divides 183, 91 or
43. Since the reminder is the same in each case,
the difference of the terms must be exactly
divisible by x. Also, x must the greatest
possible number that exactly divides the
difference between the terms. Required number, x
HCF of (183 91, 91 43, 183 43)
HCF of (92, 48, 140) 4
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