An elegant use of LCM | Kangaroo Math 2021
Updated: September 10, 2025
Summary
The video discusses a math problem involving finding positive integers less than 2021 that share the same remainder when divided by 6, 7, 8, and 9. It introduces the concept of a number leaving a remainder of five when divided by these numbers, specifically focusing on the general number N. The key idea is that N-5 must be a multiple of the least common multiple (LCM) of 6, 7, 8, and 9, which is 54. By exploring the values of k, one can find the corresponding values of N that satisfy the given property. The video provides a step-by-step approach to determine the first few values of N meeting the criteria.
Introduction to Math Kangaroo Question
A math kangaroo question involving the number 2021 and its remainder when divided by 6, 7, 8, and 9. The question asks for the number of positive integers less than 2021 that have the same remainder property.
Property of Number N
Discussing a general number N that leaves a remainder of five when divided by 6, 7, 8, and 9. The property indicates that N-5 is a multiple of 6, 7, 8, and 9.
Calculation of Least Common Multiple (LCM)
Explaining the calculation of the least common multiple (LCM) of 6, 7, 8, and 9, which is 54. This leads to the form 54 multiplied by a whole number k for N-5.
Expression for N
Showing that N-5 is of the form 54 multiplied by a whole number k. Exploring the values of K to find the corresponding values of N.
Values of N satisfying the Property
Determining the first few values of N that are smaller than 2021 and satisfy the specified property.
FAQ
Q: What does the property of a number N leaving a remainder of five when divided by 6, 7, 8, and 9 imply?
A: The property implies that N-5 is a multiple of 6, 7, 8, and 9.
Q: How is the least common multiple (LCM) of 6, 7, 8, and 9 calculated?
A: The LCM of 6, 7, 8, and 9 is calculated as 54.
Q: What is the form of the number N-5 based on the LCM calculation?
A: The form is 54 multiplied by a whole number k for N-5.
Q: How can the values of N satisfying the specified property be found?
A: By exploring the values of k and multiplying 54 with k to find the corresponding values of N.
Q: What are some of the first few values of N that are smaller than 2021 and satisfy the property?
A: The first few values include numbers such as 49, 103, 157, 211, etc., which are obtained by adding 5 to multiples of 54.
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