Find all natural numbers n that satisfy two of the given statements: 1) Number n is a perfect square.
2) The units digit of the number n is 3. 3) Number n + 15 is a perfect square.
Answers
Answer:
Step-by-step explanation:
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Given : numbers n that satisfy two of the given statements:
1) Number n is a perfect square.
2) The units digit of the number n is 3.
3) Number n + 15 is a perfect square.
To Find : all natural numbers n that satisfy two of the given statements:
Solution:
Prefect squares can ends with unit digits 0 , 1 , 4 ,5 , 6 or 9 only
Number n is a perfect square. = k²
The units digit of the number n is 3. ( Perfect square can not have unit digit 3)
so 1st and 2nd statement does not have any thing in common
n + 15 is a perfect square. Hence n + 15 = k²
Checking 1st and 3rd
n + 15 = Perfect square = b²
n = Perfect square = a²
=> a² + 15 = b²
=> (b + a) (b - a) = 15
15 = 5 x 3 or 15 x 1
=> b + a = 5 , b - a = 3 or b + a = 15 , b - a = 1
=> b = 4 , a = 1 or b = 8 , a = 7
n= 1 , n = 49
Checking 2nd and third
The units digit of the number n is 3
n + 15 is a perfect square.
3 + 15 = 18
no perfect square ends with 8
Hence not possible
So only possible values are 1 and 49
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