How many positive integers less than 1000 a) aredivisibleby7? b) are divisible by 7 but not by 11? c) are divisible by both 7 and 11? d) are divisible by either 7 or 11? e) are divisible by exactly one of 7 and 11? f) aredivisiblebyneither7nor11? g) havedistinctdigits?

Answer:a.142 b.130 c.12 d.220 e.208 f.779 g.720Step-by-step explanation:How many positive integers less than 1000 (from 1 to 999)a) are divisible by 7?999/7=142,71then there are 142 integers that are divisible by 7 7*1=77*2=147*3=21...7*141=9877*142=994 b) are divisible by 7 but not by 11?as 7 and 11 are prime numbers7*11=77999/77=12.97142-12=130c) are divisible by both 7 and 11? 999/77=12.97then there are 12 integers that are divisible by both 11 and 7 below 1000d) are divisible by either 7 or 11? 999/7=142.71999/11=90.81then the numbers that are divisible either by 7 or 11 is 142+90-12(the numbers that are counted in both grups 'divisible by 7' and 'divisible by 11')220e) are divisible by exactly one of 7 and 11? [142('divisible by 7')-12('divisible by both')]+[90('divisible by 11')-12('divisible by both')]208f) are divisible by neither 7 nor 11?999('all the positive integers below 1000)-220('are divisible by either 7 or 11')779 g) have distinct digits?For evey digit x from 0 to 90xx,1xx,2xx,3xx,4xx,5xx,6xx,7xx,8xx,9xx 10 optionsxx0,xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 10 optionsx0x,x1x,x2x,x3x,x4x,x5x,x6x,x7x,x8x,x9x 10 optionsand xxx one optionfor each of the first three rows we have to sustract the option in wich the three digits are the same digit.for the 0 we have to sustract the 000 of the fourth row as it is not a positive number. 10 (different digits)*(9+9+9+1)(different combinations for each number)-1(the 000)=279999-279=720