Important algebra identities and questions

Some Basic Formulae:

  1. (a + b)(a – b) = (a2 – b2)
  2. (a + b)2 = (a2 + b2 + 2ab)
  3. (a – b)2 = (a2 + b2 – 2ab)
  4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
  5. (a3 + b3) = (a + b)(a2 – ab + b2)
  6. (a3 – b3) = (a – b)(a2 + ab + b2)
  7. (a3 + b3 + c3 – 3abc) = (a + b + c)(a2 + b2 + c2 – ab – bc – ac)
  8. When a + b + c = 0, then a3 + b3 + c3 = 3abc.

1. If one-third of one-fourth of a number is 15, then three-tenth of that number is:

2. Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:

3. The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?

4. The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2 ?

5. A two-digit number is such that the product of the digits is 8. When 18 is added to the number, then the digits are reversed. The number is

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