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Difficulty: HardConsecutive Integers and Number Sets

A set SS consists of nn consecutive integers. The sum of all the integers in set SS is 675675, and the product of the smallest integer and the largest integer in set SS is 19761{}976. What is the value of nn?

Answer: 15

Answer

15
For any set of nn consecutive integers, the median mm equals the arithmetic mean 675n\frac{675}{n}. The smallest and largest elements can be written as m−n−12m - \frac{n-1}{2} and m+n−12m + \frac{n-1}{2}, respectively. Their product is m2−(n−12)2=1976m^2 - \left(\frac{n-1}{2}\right)^2 = 1976. Substituting n=15n = 15 gives m=45m = 45, leading to 452−72=2025−49=197645^2 - 7^2 = 2025 - 49 = 1976, which satisfies all conditions.

Step-by-Step Solution

1
Relate the sum of the set to its mean and number of terms
The mean (arithmetic average) of nn consecutive integers is equal to the median mm, so Sum=n⋅m=675\text{Sum} = n \cdot m = 675, which implies m=675nm = \frac{675}{n}.
For any evenly spaced set, the sum equals the number of terms times the mean.
2
Express the smallest and largest elements in terms of the median mm and number of terms nn
The smallest element is a=m−n−12a = m - \frac{n-1}{2} and the largest element is b=m+n−12b = m + \frac{n-1}{2}.
In a set of nn consecutive integers, the distance from the median to either endpoint is n−12\frac{n-1}{2}.
3
Formulate the product equation using the difference of squares
The product of the smallest and largest elements is a⋅b=(m−n−12)(m+n−12)=m2−(n−12)2=1976a \cdot b = \left(m - \frac{n-1}{2}\right)\left(m + \frac{n-1}{2}\right) = m^2 - \left(\frac{n-1}{2}\right)^2 = 1976.
Applying the difference of squares identity (x−y)(x+y)=x2−y2(x-y)(x+y) = x^2 - y^2 simplifies the expression.
4
Substitute m=675nm = \frac{675}{n} and solve for nn
Substituting mm yields (675n)2−(n−12)2=1976\left(\frac{675}{n}\right)^2 - \left(\frac{n-1}{2}\right)^2 = 1976. Testing odd integer factors nn of 675675:
- If n=9n = 9: m=75m = 75, product =752−42=5625−16=5609≠1976= 75^2 - 4^2 = 5625 - 16 = 5609 \neq 1976.
- If n=15n = 15: m=45m = 45, product =452−72=2025−49=1976= 45^2 - 7^2 = 2025 - 49 = 1976.
Hence, n=15n = 15.
Since nn must be a positive integer factor of 675675, checking candidate factors narrows down the unique solution.

Key Concept

Properties of consecutive integer sets: mean-median equivalence and difference of squares decomposition for endpoints.

Alternative Method

Let the set be {a,a+1,…,a+n−1}\{a, a+1, \dots, a+n-1\}. The sum is n(2a+n−1)2=675  ⟹  n(2a+n−1)=1350\frac{n(2a + n - 1)}{2} = 675 \implies n(2a + n - 1) = 1350. The product of endpoints is a(a+n−1)=1976a(a + n - 1) = 1976. Solving the system of equations for integer values of aa and nn yields a=38a = 38 and n=15n = 15.
Estimated Time:2m 30s
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Topics

Consecutive Integers and Number Sets

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