Quick Check 1 — Divisibility Rules and Sum Formulas – KLEE Arithmetic & Mental Ability Test 1
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Solved examples
Which of the following numbers is divisible by 2? (176, 221, 327, 901)
Answer: 176 — A number is divisible by 2 when its last digit is 0, 2, 4, 6 or 8.
Fill the lowest possible digit in the blank to make 80426_ divisible by 3.
Answer: 1 — Digit sum 8+0+4+2+6 = 20; the next multiple of 3 is 21, so the digit is 1.
Fill the lowest possible digit in the blank to make 4568_ divisible by 4.
Answer: 0 — The last two digits 80 are divisible by 4, so 0 works.
Which of the following numbers is divisible by 5? (15487, 45682, 95655, 12879)
Answer: 95655 — A number is divisible by 5 when its last digit is 0 or 5.
Which of the following numbers is divisible by 6? (145962, 145963, 145964, 145965)
Answer: 145962 — Divisible by 6 means even AND digit sum divisible by 3.
Which of the following numbers is divisible by 7? (171, 119, 107, 383)
Answer: 119 — 11 − 2×9 = −7, a multiple of 7, so 119 is divisible by 7.
If 91870X2 is completely divisible by 8, the smallest whole number in place of X is:
Answer: 3 — The last three digits 0X2 must be divisible by 8; 032 = 32 works, so X = 3.
Which of the following numbers is divisible by 9? (8145, 2747, 3225, 3532)
Answer: 8145 — 8+1+4+5 = 18, a multiple of 9.
Which of the following numbers is divisible by 10? (3456, 3425, 5436, 4570)
Answer: 4570 — A number is divisible by 10 only when its last digit is 0.
If 97215X6 is completely divisible by 11, the smallest whole number in place of X is:
Answer: 3 — Odd-place sum 9+2+5+6 = 22, even-place sum 7+1+X; the difference 14 − X must be 0 or a multiple of 11, so X = 3.
Which of the following numbers is divisible by 24? (210103512, 210103511, 210103510, 210103515)
Answer: 210103512 — Divisible by 24 means divisible by both 3 and 8.
Which of the following numbers is divisible by 99? (114345, 913462, 135792, 3572406)
Answer: 114345 — Divisible by 99 means divisible by both 9 and 11.
Find the difference between the largest and smallest six-digit numbers.
Answer: 8,99,999 — 9,99,999 − 1,00,000 = 8,99,999.
Find the sum of the first 35 natural numbers.
Answer: 630 — S = n(n+1)/2 = 35 × 36/2 = 630.
What is the sum of the first 20 even numbers?
Answer: 420 — S = n(n+1) = 20 × 21 = 420.
Find the sum of the odd numbers from 1 to 50.
Answer: 625 — There are 25 odd numbers, so S = n² = 25² = 625.
The sum of squares of the first ten natural numbers is:
Answer: 385 — n(n+1)(2n+1)/6 = 10 × 11 × 21/6 = 385.
Practice questions
Which of the following numbers is divisible by 2?
- A. 538
- B. 417
- C. 659
- D. 893
Answer: Option A
Explanation:
538 ends in 8, an even digit; the others end in odd digits.
Which of the following numbers is divisible by 2?
- A. 7,091
- B. 7,090
- C. 7,093
- D. 7,095
Answer: Option B
Explanation:
7,090 ends in 0, so it is even.
Which of the following numbers is NOT divisible by 2?
- A. 4,320
- B. 4,327
- C. 4,326
- D. 4,324
Answer: Option B
Explanation:
4,327 ends in 7, an odd digit, so it is not divisible by 2.
How many of the numbers 112, 345, 768, 991 and 1,004 are divisible by 2?
- A. 2
- B. 5
- C. 3
- D. 4
Answer: Option C
Explanation:
112, 768 and 1,004 end in even digits; 345 and 991 do not.
Which digit placed in the blank makes 6,47_ an odd number?
- A. 0
- B. 8
- C. 4
- D. 3
Answer: Option D
Explanation:
An odd number ends in 1, 3, 5, 7 or 9; of the options only 3 is odd.
Fill the lowest possible digit in the blank to make 5214_ divisible by 3.
- A. 1
- B. 3
- C. 2
- D. 0
Answer: Option D
Explanation:
5+2+1+4 = 12 is already a multiple of 3, so the lowest digit is 0.
Fill the lowest possible digit in the blank to make 7318_ divisible by 3.
- A. 1
- B. 3
- C. 2
- D. 5
Answer: Option C
Explanation:
7+3+1+8 = 19; the next multiple of 3 is 21, so the digit is 2.
Fill the lowest possible digit in the blank to make 2_635 divisible by 3.
- A. 1
- B. 0
- C. 4
- D. 2
Answer: Option D
Explanation:
2+6+3+5 = 16; the next multiple of 3 is 18, so the digit is 2.
Fill the LARGEST possible digit in the blank to make 4_812 divisible by 3.
- A. 8
- B. 7
- C. 9
- D. 6
Answer: Option C
Explanation:
4+8+1+2 = 15; digits keeping the sum a multiple of 3 are 0, 3, 6, 9 — the largest is 9.
Which of the following numbers is divisible by 3?
- A. 51,725
- B. 51,727
- C. 51,724
- D. 51,723
Answer: Option D
Explanation:
5+1+7+2+3 = 18, a multiple of 3.
Fill the lowest possible digit in the blank to make 3247_ divisible by 4.
- A. 4
- B. 2
- C. 6
- D. 0
Answer: Option B
Explanation:
The last two digits must be divisible by 4: 70 and 71 are not, 72 is. So the digit is 2.
Fill the lowest possible digit in the blank to make 9135_ divisible by 4.
- A. 0
- B. 6
- C. 2
- D. 4
Answer: Option C
Explanation:
50 and 51 are not multiples of 4; 52 is.
Fill the lowest possible digit in the blank to make 1063_ divisible by 4.
- A. 4
- B. 6
- C. 2
- D. 0
Answer: Option C
Explanation:
30 and 31 are not multiples of 4; 32 is.
Which of the following numbers is divisible by 4?
- A. 73,518
- B. 73,516
- C. 73,522
- D. 73,514
Answer: Option B
Explanation:
The last two digits 16 are divisible by 4.
Which of the following numbers is NOT divisible by 4?
- A. 8,138
- B. 8,140
- C. 8,144
- D. 8,136
Answer: Option A
Explanation:
The last two digits 38 are not divisible by 4 (38 ÷ 4 = 9.5).
Which of the following numbers is divisible by 5?
- A. 36,783
- B. 36,781
- C. 36,785
- D. 36,789
Answer: Option C
Explanation:
36,785 ends in 5.
Which of the following numbers is divisible by 5?
- A. 90,216
- B. 90,214
- C. 90,210
- D. 90,212
Answer: Option C
Explanation:
90,210 ends in 0.
Which of the following numbers is NOT divisible by 5?
- A. 7,552
- B. 7,560
- C. 7,550
- D. 7,555
Answer: Option A
Explanation:
7,552 ends in 2, so it is not a multiple of 5.
What is the remainder when 48,763 is divided by 5?
- A. 2
- B. 3
- C. 0
- D. 1
Answer: Option B
Explanation:
Only the last digit matters: 3 ÷ 5 leaves remainder 3.
Which digit in the blank makes 2,04_ divisible by both 2 and 5?
- A. 0
- B. 8
- C. 2
- D. 5
Answer: Option A
Explanation:
To be divisible by both 2 and 5 (that is, by 10) the last digit must be 0.
Which of the following numbers is divisible by 6?
- A. 23,156
- B. 23,154
- C. 23,155
- D. 23,157
Answer: Option B
Explanation:
23,154 is even and 2+3+1+5+4 = 15 is divisible by 3.
Which of the following numbers is divisible by 6?
- A. 81,244
- B. 81,246
- C. 81,243
- D. 81,245
Answer: Option B
Explanation:
81,246 is even and its digit sum 21 is divisible by 3.
Which of the following numbers is divisible by 3 but NOT by 6?
- A. 4,224
- B. 4,218
- C. 4,215
- D. 4,212
Answer: Option C
Explanation:
4,215 has digit sum 12 (divisible by 3) but is odd, so it is not divisible by 6.
Which of the following numbers is even but NOT divisible by 6?
- A. 7,338
- B. 7,344
- C. 7,334
- D. 7,332
Answer: Option C
Explanation:
7,334 is even but its digit sum 17 is not divisible by 3.
Fill the LARGEST digit in the blank so that 5,73_ is divisible by 6.
- A. 4
- B. 8
- C. 9
- D. 6
Answer: Option D
Explanation:
5+7+3 = 15, so the digit must be 0, 3, 6 or 9 for divisibility by 3, and even for divisibility by 2. The largest such digit is 6.
Which of the following numbers is divisible by 7?
- A. 167
- B. 161
- C. 169
- D. 163
Answer: Option B
Explanation:
16 − 2×1 = 14, a multiple of 7.
Which of the following numbers is divisible by 7?
- A. 345
- B. 341
- C. 347
- D. 343
Answer: Option D
Explanation:
34 − 2×3 = 28, a multiple of 7 (343 = 7³).
Which of the following numbers is divisible by 7?
- A. 627
- B. 637
- C. 657
- D. 647
Answer: Option B
Explanation:
63 − 2×7 = 49, a multiple of 7.
Which of the following numbers is NOT divisible by 7?
- A. 406
- B. 427
- C. 419
- D. 434
Answer: Option C
Explanation:
41 − 2×9 = 23, not a multiple of 7. (406, 427 and 434 are 7×58, 7×61 and 7×62.)
Which of the following numbers is NOT divisible by 7?
- A. 520
- B. 532
- C. 511
- D. 525
Answer: Option A
Explanation:
52 − 2×0 = 52, not a multiple of 7. (511 = 7×73, 525 = 7×75, 532 = 7×76.)
If 54321X6 is divisible by 8, the smallest whole number in place of X is:
- A. 4
- B. 1
- C. 3
- D. 2
Answer: Option C
Explanation:
The last three digits 1X6 must be divisible by 8: 106, 116, 126 fail; 136 = 8 × 17 works.
If 7342X4 is divisible by 8, the smallest whole number in place of X is:
- A. 0
- B. 4
- C. 2
- D. 1
Answer: Option C
Explanation:
The last three digits 2X4: 204 and 214 fail; 224 = 8 × 28 works.
If 4817X2 is divisible by 8, the smallest whole number in place of X is:
- A. 0
- B. 1
- C. 5
- D. 3
Answer: Option B
Explanation:
The last three digits 7X2: 702 fails; 712 = 8 × 89 works.
If 35314X8 is divisible by 8, the smallest whole number in place of X is:
- A. 2
- B. 4
- C. 1
- D. 0
Answer: Option D
Explanation:
The last three digits 4X8: 408 = 8 × 51 already works, so X = 0.
Which of the following numbers is divisible by 8?
- A. 71,268
- B. 71,256
- C. 71,252
- D. 71,260
Answer: Option B
Explanation:
The last three digits 256 = 8 × 32.
What is the remainder when 5,63,214 is divided by 8?
- A. 6
- B. 2
- C. 0
- D. 4
Answer: Option A
Explanation:
Only the last three digits matter: 214 = 8 × 26 + 6.
Which of the following numbers is divisible by 9?
- A. 7,124
- B. 7,129
- C. 7,128
- D. 7,138
Answer: Option C
Explanation:
7+1+2+8 = 18, a multiple of 9.
Find the smallest digit in the blank so that 4_731 is divisible by 9.
- A. 3
- B. 5
- C. 2
- D. 0
Answer: Option A
Explanation:
4+7+3+1 = 15; the next multiple of 9 is 18, so the digit is 3.
Find the largest digit in the blank so that 62_14 is divisible by 9.
- A. 4
- B. 8
- C. 9
- D. 5
Answer: Option D
Explanation:
6+2+1+4 = 13; to reach 18 the digit must be 5 (reaching 27 would need 14, which is not a digit).
What is the remainder when 2,34,568 is divided by 9?
- A. 8
- B. 0
- C. 2
- D. 1
Answer: Option D
Explanation:
The digit sum is 28, and 28 = 9 × 3 + 1.
Which of the following numbers is divisible by 3 but NOT by 9?
- A. 1,245
- B. 1,251
- C. 1,242
- D. 1,260
Answer: Option A
Explanation:
1+2+4+5 = 12 is a multiple of 3 but not of 9. The others have digit sum 9.
Which of the following numbers is divisible by 10?
- A. 6,788
- B. 6,780
- C. 6,801
- D. 6,785
Answer: Option B
Explanation:
Only 6,780 ends in 0.
What is the remainder when 9,347 is divided by 10?
- A. 7
- B. 0
- C. 3
- D. 4
Answer: Option A
Explanation:
The remainder on division by 10 is the last digit, 7.
How many multiples of 10 are there from 1 to 250 (inclusive)?
- A. 24
- B. 25
- C. 250
- D. 26
Answer: Option B
Explanation:
10, 20, …, 250 gives 250 ÷ 10 = 25 numbers.
Which of the following is divisible by 5 but NOT by 10?
- A. 3,455
- B. 3,460
- C. 3,450
- D. 3,470
Answer: Option A
Explanation:
3,455 ends in 5, so it is a multiple of 5 but not of 10.
What is the smallest number that must be added to 4,763 to make it divisible by 10?
- A. 37
- B. 4
- C. 7
- D. 3
Answer: Option C
Explanation:
4,763 + 7 = 4,770, which ends in 0.
If 50_37 is divisible by 11, the digit in the blank is:
- A. 2
- B. 1
- C. 6
- D. 3
Answer: Option A
Explanation:
Alternating sum 5 − 0 + d − 3 + 7 = 9 + d must be a multiple of 11, so d = 2.
If 89_53 is divisible by 11, the digit in the blank is:
- A. 7
- B. 2
- C. 5
- D. 3
Answer: Option D
Explanation:
Alternating sum 8 − 9 + d − 5 + 3 = d − 3 must be 0, so d = 3.
If 685_95 is divisible by 11, the digit in the blank is:
- A. 5
- B. 7
- C. 4
- D. 8
Answer: Option B
Explanation:
Alternating sum 6 − 8 + 5 − d + 9 − 5 = 7 − d must be 0, so d = 7.
Which of the following numbers is divisible by 11?
- A. 4,764
- B. 4,763
- C. 4,773
- D. 4,765
Answer: Option B
Explanation:
4 − 7 + 6 − 3 = 0, so 4,763 is divisible by 11 (= 11 × 433).
Which of the following numbers is divisible by 11?
- A. 91,829
- B. 91,849
- C. 91,838
- D. 91,839
Answer: Option D
Explanation:
9 − 1 + 8 − 3 + 9 = 22, a multiple of 11.
Which of the following numbers is divisible by 24?
- A. 35,712
- B. 35,728
- C. 35,716
- D. 35,720
Answer: Option A
Explanation:
712 = 8 × 89 and 3+5+7+1+2 = 18, so 35,712 is divisible by both 8 and 3. 35,720 and 35,728 pass the 8-test but fail the 3-test.
Which of the following numbers is divisible by 24?
- A. 4,148
- B. 4,152
- C. 4,138
- D. 4,160
Answer: Option B
Explanation:
152 = 8 × 19 and 4+1+5+2 = 12, so 4,152 passes both tests.
A number is divisible by 24 if it is divisible by both:
- A. 4 and 6
- B. 2 and 12
- C. 3 and 8
- D. 3 and 4
Answer: Option C
Explanation:
The two factors must be co-prime with product 24: 3 × 8. (12 is divisible by 4 and 6 but not by 24.)
If 4X1,296 is divisible by 24, the smallest digit X is:
- A. 0
- B. 1
- C. 2
- D. 5
Answer: Option C
Explanation:
296 = 8 × 37 passes the 8-test; the digit sum 22 + X must be a multiple of 3, so X = 2.
Which of the following numbers is NOT divisible by 24?
- A. 5,616
- B. 5,620
- C. 5,664
- D. 5,640
Answer: Option B
Explanation:
620 ÷ 8 = 77.5, so 5,620 fails the 8-test. The others are 24 × 234, 24 × 235 and 24 × 236.
Which of the following numbers is divisible by 99?
- A. 8,712
- B. 8,701
- C. 8,812
- D. 8,722
Answer: Option A
Explanation:
8+7+1+2 = 18 (divisible by 9) and 8 − 7 + 1 − 2 = 0 (divisible by 11).
Which of the following numbers is divisible by 99?
- A. 12,178
- B. 12,187
- C. 12,167
- D. 12,177
Answer: Option D
Explanation:
1+2+1+7+7 = 18 and 1 − 2 + 1 − 7 + 7 = 0, so it passes both tests (= 99 × 123).
Which of the following numbers is divisible by 99?
- A. 30,798
- B. 30,897
- C. 30,978
- D. 30,987
Answer: Option D
Explanation:
3+0+9+8+7 = 27 and 3 − 0 + 9 − 8 + 7 = 11, so it is divisible by 9 and 11 (= 99 × 313).
To check whether a number is divisible by 99, you test divisibility by:
- A. 9 and 3
- B. 3 and 11
- C. 9 and 11
- D. 3 and 33
Answer: Option C
Explanation:
9 and 11 are co-prime and 9 × 11 = 99.
If 5_133 is divisible by 99, the digit in the blank is:
- A. 3
- B. 5
- C. 6
- D. 9
Answer: Option C
Explanation:
Digit sum 12 + d must be a multiple of 9 and alternating sum 5 − d + 1 − 3 + 3 = 6 − d must be 0, so d = 6.
Find the difference between the largest and smallest five-digit numbers.
- A. 99,998
- B. 9,999
- C. 89,999
- D. 90,000
Answer: Option C
Explanation:
99,999 − 10,000 = 89,999.
Find the sum of the largest and smallest four-digit numbers.
- A. 10,000
- B. 10,999
- C. 10,998
- D. 11,000
Answer: Option B
Explanation:
9,999 + 1,000 = 10,999.
Find the difference between the largest four-digit number and the smallest three-digit number.
- A. 8,999
- B. 9,989
- C. 9,899
- D. 9,900
Answer: Option C
Explanation:
9,999 − 100 = 9,899.
Find the difference between the largest and smallest five-digit numbers having all digits different.
- A. 87,531
- B. 88,531
- C. 88,641
- D. 89,999
Answer: Option B
Explanation:
Largest = 98,765 and smallest = 10,234; 98,765 − 10,234 = 88,531.
How many three-digit numbers are there in all?
- A. 1,000
- B. 899
- C. 999
- D. 900
Answer: Option D
Explanation:
100 to 999 gives 999 − 100 + 1 = 900 numbers.
Find the sum of the first 50 natural numbers.
- A. 1,275
- B. 2,550
- C. 1,250
- D. 1,225
Answer: Option A
Explanation:
50 × 51/2 = 1,275.
Find the sum of the first 80 natural numbers.
- A. 6,480
- B. 3,160
- C. 3,240
- D. 3,200
Answer: Option C
Explanation:
80 × 81/2 = 3,240.
Find the sum of the natural numbers from 21 to 40.
- A. 610
- B. 630
- C. 820
- D. 600
Answer: Option A
Explanation:
Sum 1–40 = 820 and sum 1–20 = 210; 820 − 210 = 610.
Find the sum of the first 100 natural numbers.
- A. 5,050
- B. 10,100
- C. 5,000
- D. 4,950
Answer: Option A
Explanation:
100 × 101/2 = 5,050.
Find the sum of the first 45 natural numbers.
- A. 1,035
- B. 2,070
- C. 1,080
- D. 990
Answer: Option A
Explanation:
45 × 46/2 = 1,035.
What is the sum of the first 30 even numbers?
- A. 960
- B. 900
- C. 930
- D. 465
Answer: Option C
Explanation:
n(n+1) = 30 × 31 = 930.
What is the sum of the first 25 even numbers?
- A. 625
- B. 600
- C. 676
- D. 650
Answer: Option D
Explanation:
25 × 26 = 650.
Find 2 + 4 + 6 + … + 100.
- A. 2,550
- B. 5,050
- C. 2,500
- D. 2,450
Answer: Option A
Explanation:
There are 50 even numbers, so the sum is 50 × 51 = 2,550.
What is the sum of the first 15 even numbers?
- A. 240
- B. 256
- C. 120
- D. 225
Answer: Option A
Explanation:
15 × 16 = 240.
Find the sum of the even numbers from 22 to 60.
- A. 800
- B. 840
- C. 930
- D. 820
Answer: Option D
Explanation:
Sum of the first 30 evens (2 to 60) = 930; subtract the first 10 evens (2 to 20) = 110: 930 − 110 = 820.
Find the sum of the first 30 odd numbers.
- A. 930
- B. 961
- C. 870
- D. 900
Answer: Option D
Explanation:
n² = 30² = 900.
Find the sum of the odd numbers from 1 to 99.
- A. 5,000
- B. 2,550
- C. 2,500
- D. 2,450
Answer: Option C
Explanation:
There are 50 odd numbers, so the sum is 50² = 2,500.
Find the sum of the first 18 odd numbers.
- A. 361
- B. 324
- C. 342
- D. 289
Answer: Option B
Explanation:
18² = 324.
Find the sum of the odd numbers from 1 to 79.
- A. 1,560
- B. 1,600
- C. 1,521
- D. 1,640
Answer: Option B
Explanation:
1 to 79 contains 40 odd numbers, so the sum is 40² = 1,600.
Find the sum of the odd numbers from 21 to 59.
- A. 900
- B. 800
- C. 780
- D. 820
Answer: Option B
Explanation:
Odd numbers 1–59 (30 terms) sum to 900; odd numbers 1–19 (10 terms) sum to 100; 900 − 100 = 800.
The sum of squares of the first 12 natural numbers is:
- A. 600
- B. 780
- C. 506
- D. 650
Answer: Option D
Explanation:
12 × 13 × 25/6 = 650.
The sum of squares of the first 15 natural numbers is:
- A. 1,240
- B. 1,140
- C. 1,225
- D. 1,360
Answer: Option A
Explanation:
15 × 16 × 31/6 = 1,240.
The sum of squares of the first 20 natural numbers is:
- A. 2,470
- B. 2,780
- C. 3,080
- D. 2,870
Answer: Option D
Explanation:
20 × 21 × 41/6 = 2,870.
Find 11² + 12² + … + 20².
- A. 2,485
- B. 2,585
- C. 2,385
- D. 2,870
Answer: Option A
Explanation:
(1² + … + 20²) − (1² + … + 10²) = 2,870 − 385 = 2,485.
The sum of squares of the first 7 natural numbers is:
- A. 140
- B. 204
- C. 120
- D. 91
Answer: Option A
Explanation:
7 × 8 × 15/6 = 140.