What is the value of x?
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Hello!
The equation, y = mx + b is slope-intercept form. In this equation, m is the slope, and b is the y-intercept.
If the baby was exactly 8 pounds when it was born, then the y-intercept is (0, 8) because at zero months, the baby was eight pounds. To find the rate of change, we can use the the y-intercept (0, 8), and the weight of the baby at four months, which is (4, 8 + 3) → (4, 11).
Since we have two points, we can use the slope formula to find the rate of change.
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The rate of change is 3/4.
Therefore, the equation that describes the baby's weight is y = 3/4x + 8.
Since 2⁵ = 32, and
2⁵ ≡ 32 ≡ 10 (mod 22),
we have
2⁵¹ ≡ 2 • 2⁵⁰ ≡ 2 • (2⁵)¹⁰ ≡ 2 • 10¹⁰ (mod 22)
Now consider 10¹⁰ (mod 22):
10 = 2 • 5
10¹⁰ ≡ 2¹⁰ • 5¹⁰ ≡ (2⁵)² • 5¹⁰ ≡ 10² • 5¹⁰ ≡ 2² • 5¹² (mod 22)
so that
2⁵¹ ≡ 2³ • 5¹² (mod 22)
Now consider 5¹² (mod 22):
5 and 22 are coprime, and ɸ(22) = 10 (where ɸ(<em>n</em>) is the Euler totient function). By Euler's theorem,
5¹² ≡ 5² • 5¹⁰ ≡ 5² • 1 ≡ 25 ≡ 3 (mod 22)
and so
2⁵¹ ≡ 2³ • 3 ≡ 24 ≡ 2 (mod 22)
Another, more tedious method: Start with smaller powers of 2 and look for a pattern.
2 ≡ 2 (mod 22)
2² ≡ 4 (mod 22)
2³ ≡ 8 (mod 22)
2⁴ ≡ 16 (mod 22)
2⁵ ≡ 32 ≡ 10 (mod 22)
2⁶ ≡ 2 • 32 ≡ 2 • 10 ≡ 20 (mod 22)
2⁷ ≡ 2 • 20 ≡ 40 ≡ 18 (mod 22)
2⁸ ≡ 2 • 18 ≡ 36 ≡ 14 (mod 22)
2⁹ ≡ 2 • 14 ≡ 28 ≡ 6 (mod 22)
2¹⁰ ≡ 2 • 6 ≡ 12 (mod 22)
2¹¹ ≡ 2 • 12 ≡ 24 ≡ 2 (mod 22)
2¹² ≡ 2 • 2 ≡ 4 (mod 22)
and so on, with a cyclic pattern of length 10. That is, for any integer <em>k</em> ≥ 0. So 2⁵¹ ≡ 2 (mod 22).