1 · Place value to the millions
In 4,738,215 each digit is worth ten times the digit to its right. The 4 is 4 millions (4,000,000), the 7 is 7 hundred thousands (700,000), and the 3 is 3 ten thousands (30,000).
If the 3 changes to an 8, the number grows by 5 ten thousands, which is 50,000. Always ask which place the digit is in before you do anything else.
2 · Rounding
To round 46,382 to the nearest thousand, look at the thousands digit (6) and the digit just right of it (3). 3 is less than 5, so the 6 stays: 46,000.
Rounding 46,382 to the nearest hundred: the hundreds digit is 3 and the next digit is 8, so round up to 46,400.
3 · Estimating a product
To estimate 612 × 48, round each number first: 612 is about 600 and 48 is about 50. Then 600 × 50 = 30,000.
Count zeros carefully. 6 × 5 = 30, then add the three zeros from 600 and 50. Most wrong estimates are off by one zero.
4 · Order of operations
Work in this order: parentheses first, then × and ÷ from left to right, then + and − from left to right.
18 − 2 × (3 + 4): parentheses 3 + 4 = 7, then 2 × 7 = 14, then 18 − 14 = 4. Going left to right without the rules gives (18 − 2) × 7 = 112, which is wrong.
5 · Factors and multiples
A factor divides a number with no remainder. The factors of 24 come in pairs: 1 × 24, 2 × 12, 3 × 8, 4 × 6. So 24 has 8 factors.
A multiple is what you get when you multiply: the multiples of 6 are 6, 12, 18, 24, 30, and so on. Factors of a number are never bigger than it; multiples are never smaller.
6 · Prime and composite numbers
A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 … A number with more factors is composite. 1 is neither.
Some numbers only look prime. 51 = 3 × 17, 57 = 3 × 19, and 91 = 7 × 13. Test small primes (2, 3, 5, 7) before you decide.
7 · Greatest common factor (GCF)
"36 pencils and 48 erasers go into identical bags with none left over. What is the greatest number of bags?" The number of bags must divide both 36 and 48.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The greatest one they share is 12, so 12 bags, each with 3 pencils and 4 erasers.
8 · Least common multiple (LCM)
"One light blinks every 6 seconds, another every 8 seconds. They blink together now. When is the next time?" List multiples until they match.
Multiples of 6: 6, 12, 18, 24. Multiples of 8: 8, 16, 24. They meet again in 24 seconds. 6 × 8 = 48 is also a common multiple, but not the first one.
9 · Divisibility rules
2: the last digit is even. 5: the last digit is 0 or 5. 10: the last digit is 0.
3: the digits add to a multiple of 3. 9: the digits add to a multiple of 9. For 5,382: 5 + 3 + 8 + 2 = 18, so 5,382 is divisible by both 3 and 9.
10 · Remainders, worked backwards
"A number divided by 7 has a quotient of 12 and a remainder of 5. What is the number?" Undo the division: 12 groups of 7 is 84, plus the 5 left over is 89.
Check: 89 ÷ 7 = 12 R5. The remainder must always be smaller than the number you divide by.
11 · Odd and even rules
even + even = even, odd + odd = even, odd + even = odd. even × anything = even, and odd × odd = odd.
These rules answer questions without any arithmetic. "The sum of two odd numbers is multiplied by 3. Is the result odd or even?" odd + odd is even, and even × 3 is even.
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