How many natural numbers n satisfy given conditions: when 5 is added to n it gives
a perfect square and when 11 is subtracted from n it also gives a perfect square?
Answers
Answer:
2
Step-by-step explanation:
Only the two numbers 11 and 20 are capable of having perfect squares when 5 is added or 11 is subtracted from them.
Given : When 5 is added to the natural number n it gives a perfect square and when 11 is subtracted from n it also gives a perfect square.
To Find : How many natural numbers n satisfy
Solution:
5 is added to the natural number n it gives a perfect square
when 11 is subtracted from n it also gives a perfect square.
n + 5 = a²
n - 11 = b²
n + 5 - (n - 11) = a² - b²
=> 16 = a² - b²
=> 16 = (a + b)(a - b)
16 = 1 * 16 = 2 * 8 = 4 * 4
a + b = 16 a - b = 1 => a = 8.5 not an integer
a + b = 8 a - b = 2 => a = 5 b = 3
n + 5 = 5² = 25 => n = 20
n - 11 = 3² = 9 => n = 20
n = 20
a + b = 4 a - b = 4 => a = 4 b = 0
n + 5 = 4² = 16 => n = 11
n - 11 = 0² = 0 => n = 11
n = 11
Hence n = 20 or 11
two such natural numbers satisfy given conditions
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