this question is one of the toughest question which include high thinking skills so answer
615 + x^2 = 2^y
find x and y for the following
Answers
Answer:
x=59,-59
y=12
Step-by-step explanation:
615 + x2 = 2y
First we know x ≥ 0, so 615 ≤ 2y, and so y > 9 since 29 = 512.
In order for the two sides to be equal, the last digit of each side must be equal. So let’s start by analyzing the last digit of 615 + x2.
Last digit of
x – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
x2 – 0, 1, 4, 9, 6, 5, 6, 9, 4, 1
From the last calculation we can add 5 to get the last digit of 615 + x2:
Last digit of
615 + x2 – 5, 6, 9, 4, 1, 0, 1, 4, 9, 6
The powers of 2 have a pattern. The last digit cycles every 4 terms as 2, 4, 8, 6 starting with y = 1, 2, 3, 4, …
In order that the last digits are equal, we want 615 + x2 to have an even last digit. Hence we must have a last digit of 4 or 6.
The last digit of 4 or 6 only occurs when y is 2, 4, …, or when y is even.
So we have established that y is even, or equal to 2n for some integer n. How does this help us? It is actually the entire key to this problem!
We can now write:
615 + x2 = 2y
615 + x2 = 22n
615 = 22n – x2
Now we can do a neat algebra trick. The right hand side is a difference of two square numbers, so we can use the formula a2 – b2 = (a + b)(a – b). So we get:
615 = 22n – x2
615 = (2n + x)(2n – x)
We now consider the factors of 615. Since 615 = 3 × 5 × 41, there are only a few possibilities:
615 = 1 × 615
615 = 3 × 205
615 = 5 × 123
615 = 15 × 41
We can further limit the possibilities because if the two factors are (2n + x) and (2n – x), then we have
sum of factors
(2n + x) + (2n – x) = 2(2n) = 2n+1
So we need the sum of the factors to be a power of 2.
615 = 1 × 615, sum(factors) = 616
615 = 3 × 205, sum(factors) = 208
615 = 5 × 123, sum(factors) = 128
615 = 15 × 41, sum(factors) = 56
The only sum that is a power of 2 is 128. So we have 128 = 27 = 2n+1, which means n = 6 and so y = 2n = 12.
How to solve for x? An easy way is to use the difference of factors:
difference of factors
(2n + x) – (2n – x) = 2x
or
(2n – x) – (2n + x) = -2x
So we have 123 – 5 = 118 = 2x, meaning x = 59; or we have 123 – 5 = 118 = -2x, meaning x = -59.
And we’ve proven the only integer solutions are (x = 59, y = 12) and (x = -59, y = 12). We can also verify:
615 + 59^2 = 4096 = 2^12
615 + (-59)^2 = 4096 = 2^12