MSC #17: By Alternate Elimination and Retention
“What is the lowest common multiple of 38 808 and 1320? A) 64 680; B) 194 040; C) 582 120; D) 1 552 320; E) 2 134 440”
This is a question given in the intermediate level of the 2nd International Vedic Mathematics Olympiad (IVMO 2022) which was held on November 26, 2022.
The quickest solution to this problem involves finding the highest common factor or the greatest common divisor, which is the topic of the 17th of our 30 MATH-Inic Specials for Christmas.
The 17th MATH-Inic Special for Christmas is about the Vedic Math sub-Sutra, “By Alternate Elimination and Retention”. To simplify solutions of some problems, we sometimes ignore or temporarily eliminate some variables and retain only the others.
One of the most common applications of this sutra is in finding the x and y intercepts of a straight line like 2y + 3x = 18. We eliminate y by equating it to 0 and solve for x. We can also eliminate x to solve for y.
In combination with other Sutras like “By Addition and By subtraction” and “Proportionately”, we can use it to find the Highest Common Factor (or Greatest Common Divisor) of numbers and polynomials by addition or subtraction of multiples of the numbers. This method is similar but more flexible than the Euclidian algorithm used to solve this type of problem.
If we want to find the GCD of 589 and 437, we can first find their difference: 589 – 437 = 152. Now we can do one of two things: 1) we can subtract twice 152 from 437, 437 – 2 x 152 = 437 – 304 = 133 and then perform this subtraction: 152 – 133 = 19 or 2) subtract 437 from 3 times 152: 3 x 152 – 437 = 456 – 437 = 19. In both cases we obtained 19 which we can verify as the HCF of the two numbers.
We can also create zeroes at the end of the numbers if we add 3 times 437 to 589, we will get 3 x 437 + 589 = 1311 + 589 = 1900. Obviously 100 is not a factor of the 2 numbers but 19 can be.
The IVMO question can be solved the fastest way by finding the highest common factor of 38 808 and 1 320 first. We can do this by subtracting the former from 30 times the latter:
(30 x 1 320) – 38 808 = 39 600 – 38 808 = 792
This 792 also contains the HCF of the two numbers. It is easily seen if we let H be the HCF and aH = 38 808 and bH = 1 320. Thus 792 = 30bH – aH = (30b – a)H
Now we can eliminate 38 808, the biggest number in the computation and retain 1320.
1320 – 792 = 528.
We repeat the previous step, again, eliminating the biggest number and retaining the two smaller numbers until we reach the answer.
792 – 528 = 264
528 – 264 = 264.
This is the HCF of 38 808 and 1320 and we can readily see that 1320 = 264 x 5.
To find the LCM of the two numbers we just have to multiply 38 808 by 5. This can be mentally done by using the technique discussed in MSC #9: Proportionately- Inverse Proportion.
38 808 x (10/2) = 194, 040.
In our featured example we alternately eliminated the highest and lowest terms of the polynomials to get their Highest Common Factor, as shown in our infographics.
If we subtract the second polynomial from the first, we will eliminate the 2x^3 terms and get (14x2 – 21x + 7) which can be factored into 7(2x2 – 3x + 1)
And when we add thrice the first polynomial and four times the second, we will eliminate the constant terms and get (14x3 – 21x2 + 7x) which can be factored into 7x(2x2 – 3x + 1)
The HCF then is (2x2 + 3x + 1).
