Modulo Remainders Quiz: Understanding Modular Math

The largest multiple of 7 that does not exceed 83 is 7 multiplied by 11 equals 77. Subtracting gives 83 minus 77 equals 6. The remainder when 83 is divided by 7 is 6. Confirming: 7 multiplied by 11 equals 77, plus 6 equals 83. Since 6 is less than the divisor 7, this is a valid remainder.

The largest multiple of 8 that does not exceed 117 is 8 multiplied by 14 equals 112. Subtracting gives 117 minus 112 equals 5. The remainder when 117 is divided by 8 is 5. Confirming: 8 multiplied by 14 equals 112, plus 5 equals 117. Since 5 is less than the divisor 8, this is a valid remainder.

The answer is True. Since 10 leaves remainder 1 when divided by 9, raising 10 to any power raises 1 to that same power in modular arithmetic. 1 raised to any power is still 1, so 10 squared, 10 cubed, and any higher power all leave remainder 1 when divided by 9. For example, 100 divided by 9 gives remainder 1, and 1000 divided by 9 also gives remainder 1.

The largest multiple of 11 that does not exceed 256 is 11 multiplied by 23 equals 253. Subtracting gives 256 minus 253 equals 3. The remainder when 256 is divided by 11 is 3. Confirming: 11 multiplied by 23 equals 253, plus 3 equals 256. Since 3 is less than the divisor 11, this is a valid remainder.

Numbers leaving remainder 4 when divided by 9 follow the pattern 4, 13, 22, 31, 40, 49, increasing by 9 each time. Starting from 4 and listing up to 50: 4, 13, 22, 31, 40, 49 — that is exactly 6 numbers. The next in the pattern would be 58, which falls outside the range. Each of these numbers can be written as 9 multiplied by a whole number plus 4.

Two numbers are congruent modulo 6 when they leave the same remainder when divided by 6. Pair A: 14 divided by 6 leaves remainder 2 and 20 divided by 6 leaves remainder 2 — congruent. Pair B: 9 divided by 6 leaves remainder 3 and 15 divided by 6 leaves remainder 3 — congruent. Pair C: 11 divided by 6 leaves remainder 5 and 19 divided by 6 leaves remainder 1 — not congruent. Pair D: 8 divided by 6 leaves remainder 2 and 26 divided by 6 leaves remainder 2 — congruent.

The ones digit of powers of 2 repeats every 4 steps in the cycle 2, 4, 8, 6. The remainder when a power of 2 is divided by 5 follows the same period: remainders are 2, 4, 3, 1 repeating every 4 steps. To find where the 100th power falls, divide 100 by 4. 100 divided by 4 equals 25 with remainder 0. A remainder of 0 means the exponent is a multiple of 4, corresponding to the 4th position in the cycle. The 4th remainder in the cycle 2, 4, 3, 1 is 1.

The answer is False. Since 10 divided by 3 gives remainder 1, the number 10 is congruent to 1 modulo 3. Raising 10 to any power raises 1 to that same power in modular arithmetic, and 1 raised to any power remains 1. So every positive power of 10 leaves remainder 1 when divided by 3, not 2. For example, 100 divided by 3 gives remainder 1, and 1000 divided by 3 also gives remainder 1.

The multiplication property of modular arithmetic states that (a multiplied by b) mod n equals (a mod n multiplied by b mod n) mod n. Substituting the given remainders: (2 multiplied by 5) mod 7 equals 10 mod 7. Dividing 10 by 7 gives quotient 1 remainder 3. So (a multiplied by b) mod 7 equals 3.

The multiples of 7 in the range 20 to 50 are 21, 28, 35, 42, and 49. Each leaves remainder 0 when divided by 7. Counting them gives exactly 5 numbers. The next multiple of 7 after 49 would be 56, which falls outside the range. The previous multiple before 21 would be 14, which also falls outside.

The largest multiple of 13 that does not exceed 308 is 13 multiplied by 23 equals 299. Subtracting gives 308 minus 299 equals 9. The remainder when 308 is divided by 13 is 9. Confirming: 13 multiplied by 23 equals 299, plus 9 equals 308. Since 9 is less than the divisor 13, this is a valid remainder.

The largest multiple of 9 that does not exceed 203 is 9 multiplied by 22 equals 198. Subtracting gives 203 minus 198 equals 5. The remainder when 203 is divided by 9 is 5. Confirming: 9 multiplied by 22 equals 198, plus 5 equals 203. Since 5 is less than the divisor 9, this is a valid remainder.

A number is congruent to 3 mod 6 when it leaves remainder 3 when divided by 6. Checking each option: 9 divided by 6 gives quotient 1 remainder 3 — congruent. 15 divided by 6 gives quotient 2 remainder 3 — congruent. 20 divided by 6 gives quotient 3 remainder 2 — not congruent, remainder is 2 not 3. 21 divided by 6 gives quotient 3 remainder 3 — congruent. Options A, B, and D all leave remainder 3 when divided by 6.

The answer is True. This is the multiplication property of modular congruence. If a and b leave the same remainder when divided by n, then multiplying both by the same integer k preserves that congruence. For example, 11 is congruent to 3 mod 4 (both leave remainder 3). Multiplying both by 2 gives 22 and 6. 22 divided by 4 leaves remainder 2, and 6 divided by 4 also leaves remainder 2 — the congruence is preserved.

To find which locker the 127th person uses, compute 127 mod 50. 50 multiplied by 2 equals 100; 127 minus 100 equals 27. So 127 mod 50 equals 27. The 127th person uses locker 27. Confirming: the first 50 people use lockers 1 to 50, the next 50 use lockers 1 to 50 again, and the 127th person is the 27th person in the third cycle.

The 45th occurrence falls 44 intervals of 9 days after the first Monday. Total days elapsed equals 44 multiplied by 9 equals 396 days. To find the day of the week, compute 396 mod 7. 7 multiplied by 56 equals 392; 396 minus 392 equals 4. So 396 mod 7 equals 4. Starting from Monday and counting 4 days forward: Tuesday is 1, Wednesday is 2, Thursday is 3, Friday is 4. The 45th occurrence falls on a Friday.

The largest multiple of 6 that does not exceed 91 is 6 multiplied by 15 equals 90. Subtracting gives 91 minus 90 equals 1. The remainder when 91 is divided by 6 is 1. Confirming: 6 multiplied by 15 equals 90, plus 1 equals 91. Since 1 is less than the divisor 6, this is a valid remainder.

The answer is False. Modular arithmetic applies to all integers, including negative numbers. For example, negative 3 mod 5 equals 2, because negative 3 plus 5 equals 2, which is the smallest non-negative number congruent to negative 3 modulo 5. In modular arithmetic, remainders are always defined as non-negative values between 0 and n minus 1, regardless of whether the original number is negative.

Since 2 is congruent to negative 1 modulo 3 (because 2 plus 1 equals 3, which is divisible by 3), raising 2 to any even power gives negative 1 raised to that even power, which equals 1. So 2 to the power of 10 is congruent to 1 modulo 3. Confirming directly: 2 to the power of 10 equals 1024. 1024 divided by 3 gives quotient 341 remainder 1. The remainder is 1.

To find the wait time, first find how far into the current 8-minute interval the person arrives. Compute 53 mod 8: 8 multiplied by 6 equals 48; 53 minus 48 equals 5. So the person arrives 5 minutes into the current interval. Since each interval is 8 minutes long, the remaining wait time is 8 minus 5 equals 3 minutes. The next bus arrives at minute 56, and 56 minus 53 equals 3 minutes.