Question

Greatest number which divides 926 and 2313, leaving 2 and 3 remainders respectively, is?
A. 462
B. 54
C. 152
D. 154

Answer 1

Hint: The required number will divide 926 and 2313 by leaving remainders 2 and 3 respectively which means that number will indirectly divide 926+2, 2313+3. Find the highest common factor of these two resulting numbers. The H.C.F. will be the required number.

Complete step-by-step answer:
We are given that the greatest number divides 926 and 2313 leaving 2 and 3 remainders respectively.
We have to find that number.
Let the required number be ‘x’.
x leaves remainder 2 when it divides 926, this means it perfectly divides the number $926 - 2 = 924$
x leaves remainder 3 when it divides 2313, this means it perfectly divides the number $2313 - 3 = 2310$
Now to find the value of x, just find the highest common factor (H.C.F) of 924 and 2310.
To find the H.C.F of 924 and 2310, write the numbers in the product of primes and then find the common prime factors and multiply them.
924 can be written as two times 462.
$924 = 2 \times 462$
464 can be written as two times 231.
$
  462 = 2 \times 231 \\
  924 = 2 \times 2 \times 231 \\
 $
231 can be written as three times 77.
$
  231 = 3 \times 77 \\
  \therefore 924 = 2 \times 2 \times 3 \times 77 \\
 $
77 can be written as seven times 11.
$
  77 = 7 \times 11 \\
  \therefore 924 = 2 \times 2 \times 3 \times 7 \times 11 \\
 $
11 is a prime number so cannot be factored further.
2310 can be written as two times 1155.
$2310 = 2 \times 1155$
1155 can be written as three times 385.
$
  1150 = 3 \times 385 \\
  \therefore 2310 = 2 \times 3 \times 385 \\
 $
385 can be written as five times 77.
$
  385 = 5 \times 77 \\
  \therefore 2310 = 2 \times 3 \times 5 \times 77 \\
 $
77 can be written as seven times 11.
$
  77 = 7 \times 11 \\
  \therefore 2310 = 2 \times 3 \times 5 \times 7 \times 11 \\
 $
11 is a prime number so cannot be factored further.
The common factors in both the prime factorizations are 2, 3, 7 and 11.
Multiplying these factors together will give the value of x.
$
  x = 2 \times 3 \times 7 \times 11 \\
  x = 462 \\
 $
Therefore, the correct option is Option A, 462.
So, the correct answer is “Option A”.

Note: H.C.F is also known as G.C.D (Greatest common divisor). L.C.M is the least common multiple which is exactly divisible by the given numbers whereas H.C.F is the highest common factor which divides the given numbers. H.C.F and L.C.M are different. Do not confuse between them. Divides is not the same as divided by; in divides we mention the dividend and in divided by we mention the divisor i.e. dividend is divided by the divisor and divisor divides the dividend.