perplexus.info :: Numbers : Algebra busters

Algebra busters (Posted on 2020-04-22)

Find the number of solutions to the system of equations:

k₁+k₂+k₃+...+k_n=5n-4
1/k₁+1/k₂+1/k₃+...+1/k_n=1

where k₁,k₂,k₃,...,k_n and n are positive integers.

No Solution Yet Submitted by Danish Ahmed Khan

Rating: 4.0000 (1 votes)

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thoughts

| Comment 2 of 6 |

I quickly came up with three solutions:

a) k_1= 1, n=1

b) k_1 = .. = k_4 = 1, n=4

c) k_1 = 2, k_2 = 3, k_3 = 6, n=3

Let's try to constrain the k. Suppose k_1 >= k_2 >= .. >= k_n then

k_1 <= (5n-4) -(n-1)*1 = 4n-3. Also

n/k_n >= 1/k_1 + .. + 1/k_n = 1 >= n/k_1, so

k_1 >= n >= k_n

Edited on May 1, 2020, 7:36 pm

Posted by FrankM on 2020-04-26 23:15:06

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