To find: The smallest positive integer such that if we divide

avissidep 2021-09-15 Answered
To find:
The smallest positive integer such that if we divide it by three, the remainder is 2; if we divide it by five, the remainder is 3; if we divide it by seven, the remainder is 2.

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Tuthornt
Answered 2021-09-16 Author has 13696 answers
Suppose the required smallest positive integer is x.
Then by the given information, there are congruence equations
\(\displaystyle{x}\equiv{2}\pm{o}{d}{3},{x}\equiv{3}\pm{o}{d}{5},{x}\equiv{2}\pm{o}{d}{7}\).
The congruence \(\displaystyle{x}\equiv{2}\pm{o}{d}{3}\) means if x is divided by 3, the remainder is 2.
So the number x is one of the numbers from the following list:
2,5, 8, 11, 14, 17, 20, 23, 26,29,...
Similarly, the congruence \(\displaystyle{x}\equiv{3}\pm{o}{d}{5}\) means if x is divided by 5, the remainder is 3.
So the number x is one of the numbers from the following list:
3,8, 13, 18, 23, 28, 33, 38,43,...
The congruence \(\displaystyle{x}\equiv{2}\pm{o}{d}{7}\) means if x is divided by 7, the remainder is 2.
So the number x is one of the numbers from the following list:
2,9, 16,23, 30, 37,44,...
The smallest number that is found in the above three lists is 23.
So the smallest number that solves the congruences
\(\displaystyle{x}\equiv{2}\pm{o}{d}{3},{x}\equiv{3}\pm{o}{d}{5},{x}\equiv{2}\pm{o}{d}{7}\) is 23.
\(\displaystyle\Rightarrow{x}={23}\)
Final Statement:
The smallest positive integer with the given conditions is 23.
Have a similar question?
Ask An Expert
43
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-04-13
To find:
The smallest the smallest positive integer such that if we divide it by two, three and four, the remainder is one but seven divides the number evenly.
asked 2021-05-06
To find:
The smallest positive integer such that if we divide it by three, the remainder is 2. if we divide it by five, the remainder is 3. if we divide it by seven, the remainder is 2.
asked 2021-02-20
The smallest positive integer x that satisfies
\(\displaystyle{x}\equiv{3}\pm{o}{d}{5}\)
\(\displaystyle{x}\equiv{5}\pm{o}{d}{7}\)
\(\displaystyle{x}\equiv{7}\pm{o}{d}{11}\)
asked 2021-05-06
To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
\(\displaystyle{x}\equiv{1}\pm{o}{d}{\left\lbrace{4}\right\rbrace}\)
\(\displaystyle{x}\equiv{8}\pm{o}{d}{\left\lbrace{9}\right\rbrace}\)
\(\displaystyle{x}\equiv{10}\pm{o}{d}{\left\lbrace{25}\right\rbrace}\)
asked 2021-03-25
To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
\(\displaystyle{x}\equiv{1003}\pm{o}{d}{\left\lbrace{17},{369}\right\rbrace}\)
\(\displaystyle{x}\equiv{2974}\pm{o}{d}{\left\lbrace{5472}\right\rbrace}\)
asked 2021-05-04
To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
\(\displaystyle{x}\equiv{3}\pm{o}{d}{\left\lbrace{1917}\right\rbrace}\)
\(\displaystyle{x}\equiv{75}\pm{o}{d}{\left\lbrace{385}\right\rbrace}\)
asked 2021-04-19
To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
\(\displaystyle{x}\equiv{6}\pm{o}{d}{8}\)
\(\displaystyle{x}\equiv{17}\pm{o}{d}{\left\lbrace{25}\right\rbrace}\)
...