Question:medium

A number when divided by 7 leaves a remainder x. When divided by 19 it leaves a remainder 2x. Also, when divided by 39 it leaves a remainder 3x. If x = 3, then what is the least possible value of such number?

Updated On: Jul 23, 2026
  • 101
  • 211
  • 633
  • 844
  • 1266
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The Correct Option is C

Solution and Explanation

The correct answer is option (C):
633

Let the number be denoted by 'n'. We are given that:
n ≡ x (mod 7)
n ≡ 2x (mod 19)
n ≡ 3x (mod 39)

We are also given that x = 3. Substituting this value into the congruences, we have:
n ≡ 3 (mod 7)
n ≡ 2 * 3 = 6 (mod 19)
n ≡ 3 * 3 = 9 (mod 39)

From the third congruence, n ≡ 9 (mod 39), which implies that n can be written in the form n = 39k + 9 for some integer k.

Substituting this into the first congruence, we have:
39k + 9 ≡ 3 (mod 7)
Since 39 ≡ 4 (mod 7) and 9 ≡ 2 (mod 7), we get:
4k + 2 ≡ 3 (mod 7)
4k ≡ 1 (mod 7)
Multiplying by 2, we get:
8k ≡ 2 (mod 7)
Since 8 ≡ 1 (mod 7), we have:
k ≡ 2 (mod 7)
Therefore, k can be written as k = 7m + 2 for some integer m.

Substituting k = 7m + 2 into the expression for n, we get:
n = 39(7m + 2) + 9
n = 273m + 78 + 9
n = 273m + 87

Now we can verify that this satisfies the second congruence (mod 19):
n = 273m + 87
Since 273 = 14*19 + 7, 273 ≡ 7 (mod 19).
Since 87 = 4*19 + 11, 87 ≡ 11 (mod 19).
So, n ≡ 7m + 11 (mod 19).
We are given n ≡ 6 (mod 19).
Thus, 7m + 11 ≡ 6 (mod 19)
7m ≡ -5 ≡ 14 (mod 19)
Dividing by 7, which is equivalent to multiplying by the inverse of 7 (mod 19), or 11,
11 * 7m ≡ 11 * 14 (mod 19)
77m ≡ 154 (mod 19)
Since 77 = 4*19 + 1, 77 ≡ 1 (mod 19)
Since 154 = 8*19 + 2, 154 ≡ 2 (mod 19)
So we have:
m ≡ 2 (mod 19)
Thus m can be written in the form m = 19l + 2 for some integer l.

Substituting this back into the expression for n,
n = 273(19l + 2) + 87
n = 5187l + 546 + 87
n = 5187l + 633

We want the least possible positive value for n, so we choose l = 0, which yields:
n = 633

Checking the remainders:
633 divided by 7 is 90 remainder 3.
633 divided by 19 is 33 remainder 6 (which is 2 * 3).
633 divided by 39 is 16 remainder 9 (which is 3 * 3).

So the least possible value for the number is 633.
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