Praise be to Allah.
Firstly, we are happy to learn that we have a Muslim brother from Korea who is keen to understand his religion properly and to do acts of worship according to the conditions stipulated in sharee’ah. We ask Allaah to make you steadfast in adhering to His religion and to make you more keen to understand Islam properly.
With regard to the answer to your question, it varies according to how great the deviation from the qiblah is – assuming that the compass is accurate. If the deviation is great – such as if the qiblah is towards the west, and they are facing south, for example, then this deviation invalidates the prayer and the qiblah of the mosque must be changed to the right direction.
But if the deviation is slight, and is in the same direction as the qiblah – such as if the qiblah is towards the west, and they deviate slightly towards the north-west or south-west, then this does not matter and you should face this direction with them, because the ruling is that whoever is far away from the Ka’bah should face its general direction and does not have to face it precisely.
Al-Tirmidhi (344) narrated from Abu Hurayrah that the Prophet (peace and blessings of Allaah be upon him) said: “What is between the east and the west is qiblah.” (Al-Tirmidhi) said it is hasan saheeh, and it was classed as hasan by al-Albaani in his commentary on Mishkaat al-Masaabeeh (715). See also question no. 7535.
You can define the correct direction of the qiblah quite easily by looking at a map and drawing a straight line from your city to Makkah, and defining the direction of the qiblah on that basis.
And Allaah knows best.
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