Finding the Smallest Number that Leaves a Remainder of 4 When Divided by 8, 12, or 14
To solve the problem of finding the smallest number that leaves a remainder of 4 when divided by either 8, 12, or 14, we can approach it through a mathematical lens. This involves understanding the concept of congruences and the least common multiple.
Mathematical Expression and Setup
We are looking for a number n such that:
n ≡ 4 (mod 8) n ≡ 4 (mod 12) n ≡ 4 (mod 14)This means that n - 4 must be a multiple of 8, 12, and 14. In other words:
n - 4 is a multiple of 8. n - 4 is a multiple of 12. n - 4 is a multiple of 14.Calculating the Least Common Multiple (LCM)
To solve this, we first need to find the least common multiple (LCM) of 8, 12, and 14.
Prime Factorization
Let's start with the prime factorization of each number:
8 23 12 22 · 3 14 21 · 7The LCM is then determined by taking the highest power of each prime factor:
For 2: 23 (from 8) For 3: 31 (from 12) For 7: 71 (from 14)The LCM is therefore:
LCM(8, 12, 14) 23 · 31 · 71 8 · 3 · 7 168
Thus, n - 4 must be a multiple of 168.
Solving for n
Since n - 4 168k for some integer k, we can write:
n 168k 4
To find the smallest positive n, we set k 0:
n 168 · 0 4 4
However, 4 is not a valid solution because it does not leave a remainder of 4 when divided by 8, 12, or 14. Therefore, we need to try k 1:
n 168 · 1 4 172
Verification:
172 / 8 21 remainder 4 (valid) 172 / 12 14 remainder 4 (valid) 172 / 14 12 remainder 4 (valid)Hence, the smallest number that leaves a remainder of 4 when divided by 8, 12, or 14 is:
172
Conclusion
In summary, the least number that satisfies the given condition is 172. More generally, any number of the form 168n 4 (where n is a non-negative integer) will also satisfy the condition.
To make this article more SEO-friendly and Google-audible, we have discussed the mathematical logic and provided a step-by-step solution. The key points, such as the LCM, remainders, and the verification process, are highlighted to ensure clarity and understanding.