Modulo Patterns Quiz: Discover Number Remainders

The largest multiple of 8 that does not exceed 43 is 8 multiplied by 5 equals 40. Subtracting gives 43 minus 40 equals 3. The remainder when 43 is divided by 8 is 3. Confirming: 8 multiplied by 5 equals 40, plus 3 equals 43. The remainder is always less than the divisor, so a remainder of 3 when dividing by 8 is valid.

The largest multiple of 9 that does not exceed 67 is 9 multiplied by 7 equals 63. Subtracting gives 67 minus 63 equals 4. The remainder when 67 is divided by 9 is 4. Confirming: 9 multiplied by 7 equals 63, plus 4 equals 67. Since 4 is less than the divisor 9, this is a valid remainder.

The answer is True. When you divide a smaller number by a larger one, the quotient is 0 because the divisor cannot fit into the number even once. Nothing is subtracted, so the entire number remains as the remainder. For example, 3 divided by 7 gives quotient 0 and remainder 3, because 7 multiplied by 0 equals 0 and 3 minus 0 equals 3. The remainder always equals the original number when the divisor is larger.

The largest multiple of 11 that does not exceed 38 is 11 multiplied by 3 equals 33. Subtracting gives 38 minus 33 equals 5. The remainder when 38 is divided by 11 is 5. Confirming: 11 multiplied by 3 equals 33, plus 5 equals 38. Since 5 is less than the divisor 11, this is a valid remainder.

The multiples of 6 from 1 to 30 are 6, 12, 18, 24, and 30. Counting them gives exactly 5 numbers. Each leaves a remainder of 0 when divided by 6. The next multiple of 6 after 30 would be 36, which falls outside the range. Any number exactly divisible by 6 can be written as 6 multiplied by a whole number with nothing left over.

First find the remainder of 23 divided by 7: 7 multiplied by 3 equals 21; 23 minus 21 equals 2. So 23 leaves remainder 2. Checking each option: 30 divided by 7 gives quotient 4 remainder 2 — matches. 37 divided by 7 gives quotient 5 remainder 2 — matches. 44 divided by 7 gives quotient 6 remainder 2 — matches. 29 divided by 7 gives quotient 4 remainder 1 — does not match. Options A, B, and C all leave remainder 2 when divided by 7.

Days of the week repeat every 7 days, so divide 50 by 7. 50 divided by 7 equals 7 remainder 1. The remainder of 1 means 50 days from now falls 1 day beyond a complete number of weeks. Starting from Monday and moving 1 day forward gives Tuesday. The 49 complete days account for 7 full weeks and return to Monday, then 1 more day gives Tuesday.

The answer is False. A number is divisible by 3 whenever its digit sum is any multiple of 3, not only when the digit sum equals exactly 3. For example, 33 has a digit sum of 6 and is divisible by 3. The number 99 has a digit sum of 18 and is also divisible by 3. The number 12 has a digit sum of 3 and is divisible by 3. The digit sum must be a multiple of 3, which includes 3, 6, 9, 12, and so on.

The largest multiple of 6 that does not exceed 253 is 6 multiplied by 42 equals 252. Subtracting gives 253 minus 252 equals 1. The remainder when 253 is divided by 6 is 1. Confirming: 6 multiplied by 42 equals 252, plus 1 equals 253. Since 1 is less than the divisor 6, this is a valid remainder.

Adding 26 hours to 3am: 3 plus 26 equals 29. Since clocks reset every 24 hours, find the remainder when 29 is divided by 24. 29 divided by 24 equals 1 remainder 5. The remainder of 5 means the time is 5am. Confirming: adding 24 hours from 3am returns to 3am the following day, then adding the remaining 2 hours gives 5am.

Numbers leaving remainder 3 when divided by 4 follow the pattern 7, 11, 15, 19, 23, increasing by 4 each time. Starting from 4 and listing up to 25: 7, 11, 15, 19, 23 — that is exactly 5 numbers. The next in the pattern would be 27, which falls outside the range. Each of these numbers can be written as 4 multiplied by a whole number plus 3.

Option A is true: adding n to any number adds exactly one complete group of n, contributing nothing to the remainder. For example, 7 divided by 5 leaves remainder 2, and 12 divided by 5 also leaves remainder 2. Option C is true: by the definition of division, the remainder is always less than the divisor. Option B is false: doubling can push the remainder past n, causing it to wrap around. For example, 8 divided by 5 leaves remainder 3, but 16 divided by 5 leaves remainder 1, not 6. Option D is false: when the divisor exceeds the number, the remainder equals the number itself, as shown by 3 divided by 7 leaving remainder 3.

The largest multiple of 7 that does not exceed 150 is 7 multiplied by 21 equals 147. Subtracting gives 150 minus 147 equals 3. The remainder when 150 is divided by 7 is 3. Confirming: 7 multiplied by 21 equals 147, plus 3 equals 150. Since 3 is less than the divisor 7, this is a valid remainder.

The answer is True. If two numbers a and b both leave remainder r when divided by n, then a equals n multiplied by some whole number plus r, and b equals n multiplied by some other whole number plus r. Subtracting gives a minus b equals n multiplied by the difference of those whole numbers. This means n divides the difference exactly. For example, 17 and 7 both leave remainder 2 when divided by 5, and 17 minus 7 equals 10, which is exactly divisible by 5.

The ones digit of powers of 3 repeats every 4 steps in the cycle 3, 9, 7, 1. To find where the 20th power falls, divide the exponent by 4. 20 divided by 4 equals 5 with remainder 0. A remainder of 0 means the exponent is a multiple of 4, which corresponds to the 4th position in the cycle. The 4th value in the cycle 3, 9, 7, 1 is 1. So the ones digit of 3 to the power of 20 is 1.

The largest multiple of 4 that does not exceed 86 is 4 multiplied by 21 equals 84. Subtracting gives 86 minus 84 equals 2. The remainder when 86 is divided by 4 is 2. This shows that even numbers do not always leave remainder 0 when divided by 4. Only even numbers that are multiples of 4 leave remainder 0. Even numbers that are not multiples of 4 always leave remainder 2.

First find the sum: 19 plus 26 equals 45. Then divide 45 by 7. 7 multiplied by 6 equals 42; 45 minus 42 equals 3. The remainder is 3. An alternative approach uses individual remainders: 19 divided by 7 leaves remainder 5, and 26 divided by 7 also leaves remainder 5. Adding the two remainders gives 10; 10 divided by 7 leaves remainder 3. Both methods confirm the same answer.

The answer is False. Even numbers are divisible by 2 but not necessarily by 4. When any even number is divided by 4, the remainder is either 0 or 2, never anything else. For example, 8 divided by 4 gives remainder 0, but 6 divided by 4 gives remainder 2, and 86 divided by 4 gives remainder 2. Only multiples of 4 leave remainder 0. All other even numbers leave remainder 2.

The largest multiple of 8 that does not exceed 57 is 8 multiplied by 7 equals 56. Subtracting gives 57 minus 56 equals 1. The remainder when 57 is divided by 8 is 1. Confirming: 8 multiplied by 7 equals 56, plus 1 equals 57. Since 1 is less than the divisor 8, this is a valid remainder.

Numbers leaving remainder 2 when divided by 6 follow the pattern 2, 8, 14, 20, 26, 32, 38, increasing by 6 each time. Starting from 2 and listing up to 40: 2, 8, 14, 20, 26, 32, 38 — that is exactly 7 numbers. The next in the pattern would be 44, which falls outside the range. Each of these numbers can be written as 6 multiplied by a whole number plus 2.