Cracking LCM & HCF Remainder Problems: 8 Simple Formulas Explained YouTube Lecture Handouts
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Cracking LCM & HCF Remainder Problems: 8 Simple Formulas Explained
Remainder Problems with HCF and LCM
Recap
Method:
- Prime Factorization
- Divisibility tests
Problem Keywords
- LCM: Minimum number, least amount, smallest duration etc.
- HCF: Maximum number, most amount, longest duration etc.
HCF Remainder Problems
4 Types
Greatest Number Which Divides X, Y and Z? HCF Type 1 (Simple)
LetΥs start with a simple example: 12,18, and 30
Greatest Number Which Divides X, Y and Z Leaves Same Remainder R (Given) ? HCF Type 2 (Same Remainder- Given)
LetΥs find a number which divides all 14,20 and 32 leaving remainder 2
Greatest Number Which Divides X, Y and Z Leaves Same Remainder R (Not Given) ? HCF Type 3 (Same Remainder- Not Given)
Number which divides all 14,20 and 32 leaving same remainder
Greatest Number Which Divides X, Y and Z, Leaving Remainders a, B and C (Respectively) HCF Type 4 (Different Remainder- Given)
Number which divides 12,18 and 30 leaving remainder 2,3 and 0.
HCF Problems β 4 Types Summary Understand and Remember
- Greatest number which divides x, y and z = HCF (x, y, z)
- Greatest number which divides x, y and z and leaves remainder r = HCF (x - r, y - r, z - r)
- Greatest number which divides x, y and z and leaves same remainder = HCF (| x β y| , | y β z| , | z β x|)
- Greatest number which divides x, y and z and leaves remainder a, b, c = HCF (x - a, y - b, z - c)
LCM Remainder Problems
4 Types
Smallest Number Divisible by X, Y and Z? LCM Type 1 (Simple)
LetΥs start with a simple example: 6,9, and 12
Smallest/Largest Number of N Digits Divisible by X, Y, Z? LCM Type 2 (Multiples of LCM)
Smallest/Largest number of 3 digits divisible by 6,9, 12
Smallest Number when Divided by X, Y and Z Leaves Same Remainder R (Given) ? LCM Type 3 (Same Remainder)
Number divisible by 6,9, and 12 leaves remainder 2
Smallest Number when Divided by X, Y and Z Leaves Remainder a, B, C? LCM Type 4 (Different Remainder)
x - a = y - b = z - c = common difference d
Smallest number divided by 2, 3,4, 5,6 leaves remainder 1, 2,3, 4,5
LCM Problems β 4 Types SummaryUnderstand and Remember
- Smallest number divisible by x, y and z = LCM (x, y, z)
- Smallest number of n digits divisible by x, y and z = Multiple of LCM (x, y, z)
- Smallest number when divided by x, y and z leaves same remainder r = LCM (x, y, z) + r
- Smallest number when divided by x, y and z leaves remainder a, b, c
- x - a = y - b = z - c = common difference d
- LCM (a, b, c) - d
Variations of LCM (Understand)
Smallest/Largest number of n digits when divided by x, y and z leaves same remainder r = Multiple of LCM (x, y, z) + r
Smallest/Largest number of n digits when divided by x, y and z leaves remainder a, b, c = Multiple of LCM (x, y, z) β d (x-a = y-b = z-c = common difference d)
Example - 1
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.
Example - 2
Find the smallest number which, on being added 23 to it, is exactly divisible by 32,36, 48 and 96.
Example β 3 (Advanced)
When dividing a number by 12,15 or 48 there will always be a remainder of 10. If the number is the least possible, how many divisors does the number have?
Number of divisors of (p, q primes) is Application of Combination
Generalization β Chinese Remainder Theorem Next Class! !
Find the smallest number which when divided by 7,9, and 11 produces 1,2, and 3 as reminders
- 7 β 1 = 6
- 9 β 2 = 7
- 11 β 3 = 8
But
- 7 β 2 Γ 1 = 5
- 9 β 2 Γ 2 = 5
- 11 β 2 Γ 3 = 5