HCF and LCM-3 Speed Tricks & 10 Typical Word Problems YouTube Lecture Handouts

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HCF and LCM Problem

Finding our Way

Let’S Roll!

  • Some important formulas

  • Shortcut tricks!!

  • Identifying HCF Problems

  • Identifying LCM Problems

  • Looking at many different kinds of LCM and HCF problems!

  • Exercise! Its for your own good ϑ!

Recap

  • Factors (completely) divide the number. ≤ Number

  • Multiples are (completely) divisible by the number. ≥ Number

  • Highest Common Factor is the largest number which can divide all the numbers.

  • Least Common Multiple is smallest number divisible by all the numbers.

LCM and HCF Light Speed!!Use Divisibility Tests

  • 2: Even numbers

  • 3: Sum of its digits is divisible by 3.

  • 5: Number ending in 0 or 5

  • 7: Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number

  • 11: Difference of the sums of alternating digits is divisible by 11.

  • 13: Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number.

LCM and HCF Light Speed!!- Eating Small Bites

  • HCF (a, b, c) = HCF (a, HCF (b, c)) – Remember

    21, 60, 1053

  • LCM (a, b, c) = LCM (a, LCM (b, c)) – Not So Useful

  • LCM of numbers where one of them is divisible by all others will be. (2, 3, 6)

LCM and HCF Light Speed!!- Going Backwards

When numbers are divided by their HCF, the quotients have no common factors- Understand

HCF of 72 and 48 is

  • 2

  • 3

  • 6

  • 24

LCM and HCF Light Speed!!- LCM and HCF of Primes (Understand)

LCM = Product of Primes (13, 17, 11)

HCF = 1

… (1)(Understand and Remember)

  • Product of TWO numbers = Product of their H.C.F. and L.C.M.

  • The sum of two numbers is 20, their LCM is 42 and HCF is 2. The numbers are?

  • Is there a formula for 3 numbers? Yes

  • Do we need to remember that? NO

Applying Our Knowledge

Image of Applying our Knowledge

Image of Applying Our Knowledge

Image of Applying our Knowledge

If Every Wish Came True Θ

Find the highest common factor and least common multiple of the following pairs of numbers:

  • 10 and 15

  • 20 and 30

Please turn the page for solutions you can copy to your answer booklet

Example Problem - 1

3 strings of different lengths, 78 cm, 117cm and 351cm are to be cut into equal lengths. The greatest possible length of each piece is?

Image of Example of Solving of LCM - HCF

Image of Example - 1 of Solving of LCM - HCF

Image of Example of Solving of LCM - HCF

Word Problems: LCM and HCF Trigger Words Understand and Remember

LCM

  • Minimum number of days, rounds etc.

  • Least amount of time, things etc.

  • Smallest number, length, amount of time etc.

HCF

  • Maximum (possible) number of days, things etc.

  • Longest (possible) size, length, amount of time etc.

Example Problem - 2

I want to paste square pieces of equal size onto a board measuring 20 cm by 50 cm. If only whole square pieces are used, and the board is to be completely covered, find the largest possible length of the side of each square piece.

Image of Example - 2 of Solving of LCM - HCF

Image of Example - 2 of Solving of LCM - HCF

Image of Example - 2 of Solving of LCM - HCF

Example Problem - 3

A group of students can be further separated into groups of 5, 13 and 17. What is the smallest possible total number of students?

Example Problem - 4

A collection of toys can be separated into 14 equal groups in one way, and into 18 equal groups in another way. What is the smallest number of toys in the collection?

Be careful when minimum or maximum are implied!! Problem - 5

On a running track for Jay and Riya started running together, Riya completes one round in 6 minutes, while Jay takes 14 minutes. Now Jay wants to know when he will be side by side with Riya again.

Image of Cartoon For Solving LCM And HCF

Image of Cartoon for Solving LCM and HCF

Image of Cartoon For Solving LCM And HCF

Be careful when minimum or maximum are for confusing you!! Problem - 6

  • Pay close attention to the “thing” we are trying to minimize or maximize and the “thing” we are getting by LCM and HCF.

  • I want to paste square pieces of equal size onto a board measuring 20 cm by 50 cm. If only whole square pieces are used, and the board is to be completely covered, find the minimum possible number of square pieces.

Image of Square Pieces For Getting LCM and HCF

Image of Square Pieces for Getting LCM and HCF

Image of Square Pieces For Getting LCM and HCF

Let’S Understand Two Very Common Problems (7 & 8)

  • Four bells toll after interval of 8, 9, 12 and 15 minutes, respectively. If they toll together at 3 pm, when will they toll together next?

  • A small town has two bus servicing its citizens. They start simultaneously at 9 am. Bus 1 leaves the town hall every 15 mins, while bus 2 leaves every 20 mins. On a particular day, how many times did both services leave together from 9 am to 12 noon inclusive?

Grand Finale! Two Brain Teasers!! (9 & 10)

  • Riya and 3 of her friends went to a supermarket and found that a package of 6 dishcloths costs Rs. 10. If they were to share the purchase such that each has the same number of dishcloths, what is the minimum amount each has to pay?

  • Food is to be distributed to hungry college students. If there are 284 packets of biscuits, 426 packets of chocolates and 710 bottles of water, how many students are there?

In Next Class- HCF and LCM on Problems on Remainders

  • Find the greatest number of 5-digits which on being divided by 9, 12, 24 and 45 leaves 3, 6, 18 and 39 as remainders respectively.

  • Find the smallest number which, on being added 23 to it, is exactly divisible by 32, 36, 48 and 96.