Number System: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 4)

Let the original number be x. Following the operations: x + 14 = step1, step1 x 3 = step2, step2 / 2 = step3, step3 – 5 = 52. Working backward from the result 52: step3 = 52 + 5 = 57. step2 = step3 x 2 = 57 x 2 = 114. step1 = step2 / 3 = 114/3 = 38. x = step1 – 14 = 38 – 14 = 24. Check: with x=24, ((24+14) x 3)/2 – 5 = (38 x 3)/2 – 5 = 114/2 – 5 = 57 – 5 = 52, which matches.

The first set is 70, 72, 74, 76, summing to 292. The next set shifts every number forward by 8 (four steps of 2 each), so each of the 4 numbers increases by 8. Total increase = 4 x 8 = 32. New sum = 292 + 32 = 324. Check: the next set is 78, 80, 82, 84, which sums to 324.

The first set is 40, 41, 42, 43, summing to 166. The next set shifts every number forward by 4 (four steps of 1 each), so each of the 4 numbers increases by 4. Total increase = 4 x 4 = 16. New sum = 166 + 16 = 182. Check: the next set is 44, 45, 46, 47, which sums to 182.

The first set is 16, 18, 20, 22, summing to 76. The next set shifts every number forward by 8 (four steps of 2 each), so each of the 4 numbers increases by 8. Total increase = 4 x 8 = 32. New sum = 76 + 32 = 108. Check: the next set is 24, 26, 28, 30, which sums to 108.

The first set is 50, 52, 54, 56, summing to 212. The next set shifts every number forward by 8 (four steps of 2 each), so each of the 4 numbers increases by 8. Total increase = 4 x 8 = 32. New sum = 212 + 32 = 244. Check: the next set is 58, 60, 62, 64, which sums to 244.

The first set is 21, 22, 23, 24, summing to 90. The next set shifts every number forward by 4 (four steps of 1 each), so each of the 4 numbers increases by 4. Total increase = 4 x 4 = 16. New sum = 90 + 16 = 106. Check: the next set is 25, 26, 27, 28, which sums to 106.

The first set is 12, 14, 16, 18, summing to 60. The next set shifts every number forward by 8 (four steps of 2 each), so each of the 4 numbers increases by 8. Total increase = 4 x 8 = 32. New sum = 60 + 32 = 92. Check: the next set is 20, 22, 24, 26, which sums to 92.

The first set is 63, 65, 67, 69, summing to 264. The next set shifts every number forward by 8 (four steps of 2 each), so each of the 4 numbers increases by 8. Total increase = 4 x 8 = 32. New sum = 264 + 32 = 296. Check: the next set is 71, 73, 75, 77, which sums to 296.

The first set is 42, 43, 44, 45, summing to 174. The next set shifts every number forward by 4 (four steps of 1 each), so each of the 4 numbers increases by 4. Total increase = 4 x 4 = 16. New sum = 174 + 16 = 190. Check: the next set is 46, 47, 48, 49, which sums to 190.

We need primes p with 200 <= p <= 240, where a prime has no divisors other than 1 and itself. Checking each integer in this range: the primes are 211, 223, 227, 229, 233, 239. Counting these gives 6 primes. So the answer is 6.

We need primes p with 50 <= p <= 80, where a prime has no divisors other than 1 and itself. Checking each integer in this range: the primes are 53, 59, 61, 67, 71, 73, 79. Counting these gives 7 primes. So the answer is 7.

We need primes p with 160 <= p <= 200, where a prime has no divisors other than 1 and itself. Checking each integer in this range: the primes are 163, 167, 173, 179, 181, 191, 193, 197, 199. Counting these gives 9 primes. So the answer is 9.

We need primes p with 130 <= p <= 170, where a prime has no divisors other than 1 and itself. Checking each integer in this range: the primes are 131, 137, 139, 149, 151, 157, 163, 167. Counting these gives 8 primes. So the answer is 8.

We need primes p with 250 <= p <= 290, where a prime has no divisors other than 1 and itself. Checking each integer in this range: the primes are 251, 257, 263, 269, 271, 277, 281, 283. Counting these gives 8 primes. So the answer is 8.

We need primes p with 130 <= p <= 180, where a prime has no divisors other than 1 and itself. Checking each integer in this range: the primes are 131, 137, 139, 149, 151, 157, 163, 167, 173, 179. Counting these gives 10 primes. So the answer is 10.

We need primes p with 250 <= p <= 300, where a prime has no divisors other than 1 and itself. Checking each integer in this range: the primes are 251, 257, 263, 269, 271, 277, 281, 283, 293. Counting these gives 9 primes. So the answer is 9.

We need primes p with 160 <= p <= 210, where a prime has no divisors other than 1 and itself. Checking each integer in this range: the primes are 163, 167, 173, 179, 181, 191, 193, 197, 199. Counting these gives 9 primes. So the answer is 9.

We need perfect squares strictly between 345 and 527. Checking each integer’s square in that range: 361, 400, 441, 484 all fall strictly between 345 and 527. Counting these gives 4 perfect squares. So the count is 4.

We need perfect squares strictly between 305 and 529. Checking each integer’s square in that range: 324, 361, 400, 441, 484 all fall strictly between 305 and 529. Adding them: 324 + 361 + 400 + 441 + 484 = 2010. So the sum is 2010.

We need perfect squares strictly between 433 and 662. Checking each integer’s square in that range: 441, 484, 529, 576, 625 all fall strictly between 433 and 662. Adding them: 441 + 484 + 529 + 576 + 625 = 2655. So the sum is 2655.

We need perfect squares strictly between 98 and 443. Checking each integer’s square in that range: 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441 all fall strictly between 98 and 443. Adding them: 100 + 121 + 144 + 169 + 196 + 225 + 256 + 289 + 324 + 361 + 400 + 441 = 3026. So the sum is 3026.

We need perfect squares strictly between 60 and 243. Checking each integer’s square in that range: 64, 81, 100, 121, 144, 169, 196, 225 all fall strictly between 60 and 243. Counting these gives 8 perfect squares. So the count is 8.

We need perfect squares strictly between 191 and 486. Checking each integer’s square in that range: 196, 225, 256, 289, 324, 361, 400, 441, 484 all fall strictly between 191 and 486. Adding them: 196 + 225 + 256 + 289 + 324 + 361 + 400 + 441 + 484 = 2976. So the sum is 2976.

We need perfect squares strictly between 79 and 263. Checking each integer’s square in that range: 81, 100, 121, 144, 169, 196, 225, 256 all fall strictly between 79 and 263. Counting these gives 8 perfect squares. So the count is 8.

We need perfect squares strictly between 180 and 372. Checking each integer’s square in that range: 196, 225, 256, 289, 324, 361 all fall strictly between 180 and 372. Adding them: 196 + 225 + 256 + 289 + 324 + 361 = 1651. So the sum is 1651.