A complex heat source apparatus comprising: first and second, i.e., a total of two, heat exchangers (1 1, 1 2); a first burner (2 1) for heating the first heat exchanger (1 1); a second burner (2 2) for heating the second heat exchanger (1 2) in which the first burner (2 1) is larger in rated amount of combustion than the second burner (2 2), both the first and the second burners (2 1, 2 2) being constituted by totally aerated combustion burners; first and second combustion boxes (31, 32) that enclose a combustion space at the lower side of each of the first and second burners (21, 22), the combustion boxes being disposed so as to house therein the first and second heat exchanger (1 1, 1 2) respectively a single fan for supplying air-fuel mixture or primary air, the fan being interposed in a common air-fuel supply passage connected to both the first and the second burners, a downstream end of a gas supply passage for supplying fuel gas being connected to that portion of the air-fuel supply passage which is on an upstream side or a downstream side of the fan, whereby a mixture of the primary air and the fuel gas is supplied through the air-fuel supply passage to both the first and the second burners (2 1, 2 2), a flow control means interposed in the gas supply passage, said flow control means serving to vary the amount of fuel gas supply in proportion to the amount of primary air supply so that the air-fuel ratio of the air-fuel mixture becomes constant, a first on-off valve (8 1) disposed at a first connection portion (52 1) for connecting the air-fuel supply passage and the first burner (2 1) and a second on-off valve (8 2) disposed at a second connection portion (52 2) for connecting the air-fuel supply passage and the second burner (2 2) wherein the apparatus is adapted such that, at a time of simultaneous operation in which both the first and the second burners (2 1, 2 2) are combusted, both the first and the second on-off valves (8 1, 8 2) are opened and that, at a time of single operation in which only one of the first and the second burners (2 1, 2 2) is combusted, only one of the first and the second on-off valves (8 1, 8 2) corresponding to said one burner is opened; and that a flow resistance of the air-fuel mixture at the second connection portion (52 2) is set such that, in a state in which the amount of air-fuel mixture supply to the first burner (2 1), at the time of simultaneous operation, becomes a maximum amount corresponding to the rated combustion amount of the first burner (2 1), the amount of air-fuel mixture supply to the second burner (2 2) becomes smaller than the maximum amount corresponding to the rated combustion amount of the second burner (2 2). first and second, i.e., a total of two, heat exchangers (1 1, 1 2); a first burner (2 1) for heating the first heat exchanger (1 1); a second burner (2 2) for heating the second heat exchanger (1 2) in which the first burner (2 1) is larger in rated amount of combustion than the second burner (2 2), both the first and the second burners (2 1, 2 2) being constituted by totally aerated combustion burners; first and second combustion boxes (31, 32) that enclose a combustion space at the lower side of each of the first and second burners (21, 22), the combustion boxes being disposed so as to house therein the first and second heat exchanger (1 1, 1 2) respectively a single fan for supplying air-fuel mixture or primary air, the fan being interposed in a common air-fuel supply passage connected to both the first and the second burners, a downstream end of a gas supply passage for supplying fuel gas being connected to that portion of the air-fuel supply passage which is on an upstream side or a downstream side of the fan, whereby a mixture of the primary air and the fuel gas is supplied through the air-fuel supply passage to both the first and the second burners (2 1, 2 2), a flow control means interposed in the gas supply passage, said flow control means serving to vary the amount of fuel gas supply in proportion to the amount of primary air supply so that the air-fuel ratio of the air-fuel mixture becomes constant, a first on-off valve (8 1) disposed at a first connection portion (52 1) for connecting the air-fuel supply passage and the first burner (2 1) and a second on-off valve (8 2) disposed at a second connection portion (52 2) for connecting the air-fuel supply passage and the second burner (2 2) wherein the apparatus is adapted such that, at a time of simultaneous operation in which both the first and the second burners (2 1, 2 2) are combusted, both the first and the second on-off valves (8 1, 8 2) are opened and that, at a time of single operation in which only one of the first and the second burners (2 1, 2 2) is combusted, only one of the first and the second on-off valves (8 1, 8 2) corresponding to said one burner is opened; and that a flow resistance of the air-fuel mixture at the second connection portion (52 2) is set such that, in a state in which the amount of air-fuel mixture supply to the first burner (2 1), at the time of simultaneous operation, becomes a maximum amount corresponding to the rated combustion amount of the first burner (2 1), the amount of air-fuel mixture supply to the second burner (2 2) becomes smaller than the maximum amount corresponding to the rated combustion amount of the second burner (2 2).