math

posted by .

a and b are consecutive, positive integers such that a2−b2>22. What is the minimum possible value of a+b?

  • Algebra/Number Theory -

    Solution 1: Since a2>b2+22, thus a>b. Since a and b are consecutive positive integers, we have a=b+1. Substituting in the above expression, we have (b+1)2=b2+2b+1>b2+22⇒b>22−12⇒b≥11. Thus a+b=2b+1≥2×11+1=23.

    Solution 2: Since a2>b2+22, thus a>b which implies that a=b+1,a−b=1. Hence a+b=a2−b2a−b>221 ⇒a+b≥23 (since they are integers). We verify that a=11+1=12,b=11 satisfies the inequality, so the minimum possible value of a+b is indeed 23.

  • math -

    23

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. math, algebra

    2a+2ab+2b I need a lot of help in this one. it says find two consecutive positive integers such that the sum of their square is 85. how would i do this one i have no clue i know what are positive integers.but i don't know how to figure …
  2. math

    there are three consecutive positive integers such that the sum of the squares of the smallest two is 221. write and equation to find the three consecutive positive integers let x= the smallest integer
  3. Maths

    a and b are consecutive, positive integers such that a^2−b^2>22. What is the minimum possible value of a+b?
  4. MAths

    4 distinct integers p, q, r and s are chosen from the set {1,2,3,…,16,17}. The minimum possible value of p/q+r/s can be written as a/b, where a and b are positive, coprime integers. What is the value of a+b?
  5. math

    a,b and c are positive integers such that the simultaneous equations (a−2b)x=1, (b−2c)x=1 and x+25=c have a positive solution for x. What is the minimum value of a?
  6. Maths

    a and b are consecutive, positive integers such that (a+b)*(a-b)>22. What is the minimum possible value of a+b?
  7. Math (algebra)

    Suppose a and b are positive integers satisfying 1≤a≤31, 1≤b≤31 such that the polynomial P(x)=x^3−ax^2+a^2b^3x+9a^2b^2 has roots r, s, and t. Given that there exists a positive integer k such that (r+s)(s+t)(r+t)=k^2, …
  8. Maths

    Suppose a and b are positive integers satisfying 1≤a≤31, 1≤b≤31 such that the polynomial P(x)=x3−ax2+a2b3x+9a2b2 has roots r, s, and t. Given that there exists a positive integer k such that (r+s)(s+t)(r+t)=k2, …
  9. algebra!!!! please help me!!!!

    The smallest possible positive value of 1−[(1/w)+(1/x)+(1/y)+(1/z)] where w, x, y, z are odd positive integers, has the form a/b, where a,b are coprime positive integers. Find a+b.
  10. Math

    If the product of 4 consecutive integers is equal to one of the integers, what is the largest possible value of one of the integers?

More Similar Questions