Final answer:
The Muslim month for pilgrimage to Mecca is Dhu al-Hijjah, during which Muslims perform the Hajj, one of the Five Pillars of Islam.
Step-by-step explanation:
The Muslim month during which people go for pilgrimage to Mecca is Dhu al-Hijjah.
The pilgrimage, known as the Hajj, is one of the Five Pillars of Islam and is a religious duty that Muslims are expected to carry out at least once in their lifetime, if they are physically and financially able to do so. The city of Mecca holds great significance in Islam as it is the birthplace of the Prophet Muhammad and home to the Kaaba, which Muslims aim to circle during their pilgrimage. Completing the Hajj involves partaking in several rituals that reinforce the faith and unity among the community of Muslims around the world.