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Rudolph W. Giuliani weighed in on the question of Muslims cheering in America after the attacks of Sept. 11, 2001, and concluded that while he recalled some pockets of celebration in New York City, Donald J. Trump’s claim about thousands of revelers in New Jersey was an exaggeration.

The former New York mayor estimated that there were as many as 40 people who celebrated in Queens and Brooklyn after the attacks. He recalled some Muslim children in a candy store celebrating after the World Trade Center’s twin towers fell, and said that another group of children from a nearby housing development came over and beat them up.

“We did have some reports of celebrating that day while the towers were coming down,” Mr. Giuliani told CNN on Tuesday, noting that if thousands of people were cheering he would have “known that for sure.”

Mr. Trump, the Republican presidential candidate who is leading in most polls, has raised new questions about his honesty by insisting that he saw thousands of Muslims cheering in Jersey City after the attacks. He said that he watched them on television and read about such episodes in The Washington Post, although those reports have not been corroborated.

Discussing the issue last week on MSNBC, Mr. Giuliani, the New York mayor from 1994 to the end of 2001, who unsuccessfully ran for the 2008 Republican presidential nomination, said he had heard some reports from the police of such celebrations at the time but that he never saw any celebrations firsthand.

Mr. Trump has declined to retract his assertion, explaining that many people have told him that they also saw throngs of Muslims cheering from New Jersey as the towers burned and fell.