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France's iconic Eiffel Tower has plunged into darkness to honour the victims of Islamic State group bomb attacks in the Iraqi capital Baghdad.

The lights on the famous Paris monument went off late on Tuesday to commemorate the at least 42 people killed in suicide bombings the country.

"I will turn my lights off tonight, at midnight, to pay tribute to the victims of the Baghdad attacks," the official Twitter account for monument .

The first IS-claimed car bombing struck a Baghdad ice-cream shop early on Monday during the Muslim holy month of Ramadan, when Iraqis often stay out late shopping or socialising after breaking their fast.

Among the victims was 12-year-old Iraqi-Australian Zynab al-Harbiya, who was visiting her sick grandfather in Baghdad.

A second attack claimed by the group targeted the country's main pension office, while a third bombing was carried out at an army checkpoint in Hit, some 120 miles west of Baghdad.

Images posted on social media showed the devastating impact of the attack, which left ice-cream cups scattered on the bloodstained ground.

IS issued a statement claiming the first attack, identifying the bomber as an Iraqi who blew up an explosives-rigged vehicle targeting Shia Muslims.

IS considers members of Iraq's Shia Muslim majority - including the women and children who were victims of the ice-cream shop blast - to be heretics and legitimate targets for attack.

The bombings, which come just a few days into Ramadan, come as Iraqi forces fight to retake the last IS-held areas of Mosul, a city that was the extremists' emblematic stronghold.

Iraqi forces are more than seven months into a massive operation to recapture the city, and have already taken back its whole eastern side and much of the west.

Three neighbourhoods north of Mosul's Old City - al-Shifaa, al-Saha and al-Zinjili - are now the target of a broad assault by Iraqi soldiers, police and special forces that was launched last week.