The numbers 1 3 6 7 7 7 are used to form three 2 digit numbers If the sum of

# The numbers 1 3 6 7 7 7 are used to form three 2

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The numbers {1, 3, 6, 7, 7, 7} are used to form three 2-digit numbers. If the sum of these three numbers is a prime number p, what is the largest possible value of p?A. 97B. 151C. 209D. 211E. 219What is the largest possible sum of these three numbers that we can form? Maximize the first digit: 76+73+71=220=even, so not a prime. Let's try next largest sum, switch digits in 76 and we'll get: 67+73+71=211. Question is it a prime number? If you notice 210=2*3*5*7=the product of the first four primes. So, 210+1=211 must be a prime. For example: 2+1=3=prime, 2*3+1=7=prime, 2*3*5+1=31=prime.If and , what is the least value of possible?To get the least value of , which obviously will be negative, try to maximize absolute value of , as more isthe absolute value of a negative number "more" negative it is (the smallest it is).To maximize pick largest absolute values possible for and : . Notice that: -1/18<-1/36<-1/50<-1/100, so -1/100 is the largest number and -1/18 is the smallest number (we cannot obtain -1/6 from x^2*y or else it would be the correct answer).A certain bakery ran a promotion code: a customer can buy x donuts for the regular price of \$15 total and get 3 donuts free. If the donut price per dozen during the promotion is \$2 less than the normal donut price per dozen, what is x?The promotion price per donut is , thus the promotion price per dozen is ;The regular price per donut is , thus the regular price per dozen is ;
We are told that the promotion price per dozen is \$2 less than the normal donut price per dozen: --> .At this point it's better to substitute the values rather than solve for x. Answer choice A, 15, works.If , then y is NOT divisible by which of the following?Now, if you analyze each option you'll see that only is not a factor of , since the power of 13 in it is higherthan the power of 13 in .For the past k days the average (arithmetic mean) cupcakes per day that Liv baked was 55. Today Bibi joined and together with Liv they baked 100 cupcakes, which raises the average to 60 cupcakes per day. What is the value of k?A. 6B. 8C. 9D. 10E. 12Answer: D.Total cupcakes for k days was 55k, which means that total cupcakes for k+1 days was 55k+100. The new average is (55k+100)/(k+1)=60 --> 55k+100=60k+60 --> k=8What is the smallest positive integer such that is the square of a positive integer?A. 14B. 36C. 144D. 196E. 441, so in order to be a square of an integer must complete the powers of 2 and 7 to even number, so the least value of must equal to 2*7=14, which makes the leas value of equal to 14^2=196.There are 7 red and 5 blue marbles in a jar. In how many ways 8 marbles can be selected from the jar so that at least one red marble and at least one blue marble to remain in the jar?A. 460B. 490C. 493D. 455E. 445
Total ways to select 8 marbles out of 7+5=12 is ;Ways to select 8 marbles so that zero red marbles is left in the jar is ;Ways to select 8 marbles so that zero blue marbles is left in the jar is ;Hence ways to select 8 marbles so that at least one red marble and at least one blue marble to remain the jar is .