Question

Find the largest number that divides 2053 and 967 leaves a remainder of 5 and 7 respectively.

Answer 1

Hint: Firstly subtract the remainders from the number so that actual numbers and can be obtained and then calculate their factors. So the common factors among both will be our required answer.

Complete step-by-step answer:
Given 2053 and 967
We need to find the largest number that divides 2053 and 967 leaves a remainder of 5 and 7 respectively.
Calculating the actual numbers by subtracting the remainder from it as,
\[
  {\text{2053 - 5 = 2048}} \\
  {\text{967 - 7 = 960}} \\
  \]
Than calculating their factors,
\[
  {{Factors of\;2048 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \;}} \\
  {{Factors of\;960 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5}} \\
  {{HCF of\;2048\;and\;960 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64}} \\
  \]
Hence , \[64\] is our required correct answer.

Note: In arithmetic, the remainder is the integer or number that is "left over" after dividing one integer or number by another to produce an integer or number quotient . In the algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another.
In Mathematics, factorisation is defined as the break or decomposition of an entity into a product of another entity, or factors, which when multiplied together give the original number or a matrix, etc.
The Remainder Theorem: When we divide a polynomial f(x) by \[x - c\] the remainder is f(c)