Average scores on the Scholastic Aptitude Test rose last year for public high school seniors in Montgomery County and the District of Columbia, but fell slightly in Prince George's County, according to figures released yesterday.

The gains in Montgomery were the most substantial in more than a decade and occurred on both the verbal and mathematics parts of the widely used college entrance exam. District students gained five points on the math part of the exam but were unchanged in verbal. In Prince George's there was a three-point drop on the verbal portion of the test and a one-point drop in math.

The test results were reported by the College Board, which sponsors the exam, and released by the three local school systems. Spokesmen for other school districts in the area said yesterday that they expect to release their SAT scores soon.

The College Board announced last week that the nationwide average score on the SAT remained unchanged last spring, rising one point on the math part of the exam to 468, but dropping one point in verbal to 425. In 1982, scores on both parts of the exam increased for the first time in 19 years.

Last spring's eight-point SAT gain in Montgomery followed three years of steady scores on the math part of the exam and a one-point loss in 1982 in verbal. It brought the county's math average to 508, up from 503 in 1982, and its verbal average to 460, up from 457 in 1982. Possible scores on each part of the two-hour multiple-choice exam range from 200 to 800 points.

For District schools, the average scores were 368 last spring in math and 334 in verbal, a gain over the past two years of nine points in math and 11 points in verbal. The average score for D.C. public school seniors remains in the bottom 20 percent in verbal and bottom 21 percent in math of the approximately 1 million students who took the test nationwide.

The average scores for Montgomery were at the 61st percentile nationwide, meaning that the average senior in the county did as well or better than 61 percent of those taking the test around the country.

Average scores for Prince George's were 400 in verbal and 448 in math, placing its students at the 40th percentile in verbal and the 44th percentile in math. Despite this year's decline, the combined average score for Prince George's, 848, was still two points above the county's low reached in 1980.

The math portion of the SAT exam concentrates on problem solving, using arithmetic reasoning, algebra and geometry. The verbal part of the test measures reading comprehension and vocabulary.

In releasing last spring's results, the Montgomery County school board included average scores for individual high schools, while the District and Prince George's only issued system-wide averages.

Walt Whitman High School in Bethesda maintained its longtime position as Montgomery's top-scoring school, with a combined total average score of 1038 points. But Whitman's scores were down 23 points in verbal and six points in math. Its margin over the county's second highest scoring school, Winston Churchill Senior High in Potomac, narrowed to one point.

Montgomery Blair High in Silver Spring had the lowest SAT averages in the county--392 in verbal and 431 in math. Its combined average score fell 22 points--five points in verbal and 17 points in math.

James Myerberg, coordinator of testing for Montgomery County schools, said all the increase in the countywide SAT averages was accounted for by female students. The average for females on the verbal test rose from 452 in 1982 to 459 last spring, while males dropped from 462 to 461. On the math part of the exam, the average scores of female students rose ten points to 485 while the average for males was unchanged at 533.

Overall, the percentage of Montgomery students with SAT scores over 600 points rose to a 10-year high of 27 percent in the mathematics part of the test and stayed at 13 percent in verbal, where it has been since 1979. In the District, only 3.4 percent of those tested scored more than 600 points in verbal and 4.6 percent scored that high in math.