From the hadith in Sahih Al-Bukhari quoted below, we can see that 'Amr ibn Salama lead the prayers when he was 6-7 years old. If a child could lead the prayers, and it counts, then he could definitely pray behind you in a jama'ah (congregational) prayers. The condition is that said child should be able to pray properly and to actively participate in the prayer as would any adult. Refer to Islam Q&A Fatwa 155061 (in Arabic only) for more information.
NOTE — For Jum'ah (Friday) prayers, three or more are required for it to be valid (opinion of scholars). So this answer is addressing only regular prayers other than Jum'ah prayers.
حدثنا سليمان بن حرب، حدثنا حماد بن زيد، عن أيوب، عن أبي قلابة، عن عمرو بن سلمة، قال
قال لي أبو قلابة: ألا تلقاه فتسأله؟ قال فلقيته فسألته فقال: كنا بماء ممر الناس، وكان يمر بنا الركبان فنسألهم: ما للناس، ما للناس؟ ما هذا الرجل؟ فيقولون: يزعم أن الله أرسله، أوحى إليه، أو: أوحى الله بكذا
Narrated 'Amr bin Salama: We were at a place which was a thoroughfare for the people, and the caravans used to pass by us and we would ask them, "What is wrong with the people? What is wrong with the people? Who is that man?." They would say, "That man claims that Allah has sent him (as an Apostle), that he has been divinely inspired, that Allah has revealed to him such-and-such."
فكنت أحفظ ذلك الكلام، وكأنما يقر في صدري، وكانت العرب تلوم بإسلامهم الفتح، فيقولون: اتركوه وقومه، فإنه إن ظهر عليهم فهو نبي صادق، فلما كانت وقعة أهل الفتح، بادر كل قوم بإسلامهم، وبدر أبي قومي بإسلامهم، فلما قدم قال: جئتكم والله من عند النبي صلى الله عليه وسلم حقا، فقال: «صلوا صلاة كذا في حين كذا، وصلوا صلاة كذا في حين كذا، فإذا حضرت الصلاة فليؤذن أحدكم، وليؤمكم أكثركم قرآنا»
I used to memorize that (Divine) Talk, and feel as if it was inculcated in my chest (i.e. mind) And the 'Arabs (other than Quraish) delayed their conversion to Islam till the Conquest (of Mecca). They used to say." "Leave him (i.e. Muhammad) and his people Quraish: if he overpowers them then he is a true Prophet. So, when Mecca was conquered, then every tribe rushed to embrace Islam, and my father hurried to embrace Islam before (the other members of) my tribe. When my father returned (from the Prophet) to his tribe, he said, "By Allah, I have come to you from the Prophet (ﷺ) for sure!" The Prophet (ﷺ) afterwards said to them, 'Offer such-and-such prayer at such-and-such time, and when the time for the prayer becomes due, then one of you should pronounce the Adhan (for the prayer), and let the one amongst you who knows Qur'an most should, lead the prayer."
فنظروا فلم يكن أحد أكثر قرآنا مني، لما كنت أتلقى من الركبان، فقدموني بين أيديهم، وأنا ابن ست أو سبع سنين، وكانت علي بردة، كنت إذا سجدت تقلصت عني، فقالت امرأة من الحي: ألا تغطوا عنا است قارئكم؟ فاشتروا فقطعوا لي قميصا، فما فرحت بشيء فرحي بذلك القميص
So they looked for such a person and found none who knew more Qur'an than I because of the Qur'anic material which I used to learn from the caravans. They therefore made me their Imam ((to lead the prayer) and at that time I was a boy of six or seven years, wearing a Burda (i.e. a black square garment) proved to be very short for me (and my body became partly naked). A lady from the tribe said, "Won't you cover the anus of your reciter for us?" So they bought (a piece of cloth) and made a shirt for me. I had never been so happy with anything before as I was with that shirt.
In concluding Friday prayers, puberty, intellect and masculinity are required, and the minimum number required to perform Friday prayers is five or seven people, one of whom must be an Imam and is performed only by men, although women can also participate. Therefore, if the believers are less than the permissible limit, Friday prayers will not be held in the presence of distinguished women and children.